There are two core build options for going AD: one with Nashor's Tooth and one Without. Going without it (or replacing it with Blade of the Ruined King) has the benefit of slightly higher single-target DPS, but the drawback is your cleaves deal less damage, and overall less of your damage will be magic damage, meaning armor makes a bigger difference, and most champs will build armor before MR in most games. Note: that armor makes less of a difference after level 16 since your magic damage cleaves turn into true damage cleaves.

The general build itself was determined by Kayle's damage distribution (see later chapters), but the order was based on my personal preference of laning style. I'm more of a no-nonsense, no-risks kinda guy when I'm playing champs that do poorly in lane, so I went with Essence Reaver as my first item, since it lends itself to poking more, and it also complements the general goal of the build, which is attack speed and crit chance. If you prefer to get up in your opponent's face, or if your jungler is camping your lane, then Nashor's Tooth might be a better item to rush since it deals a little more consistent damage, although you will probably struggle a little before level 6, since you'd be skipping Caulfield's Warhammer. Nashor's Tooth is just an all-around good item on Kayle, and you will be able to complete it either right before or shortly after level 11, which is when attack speed really starts to matter, hence why it is her second core damage item. Infinity Edge vs Runaan's Hurricane as the 3rd item (excluding boots) is just a preference choice to me. I feel better with IE, some people prefer having loaded attack speed. The order might also depend on what you do for situational items.

General guide to situational choices

Guinsoo's Rageblade: This will round out your hybrid DPS, so I recommend saving it for your last item in situations where you aren't dying very often. Using either core AD build, this item will provide you with one of Kayle's highest single-target DPS potentials, if not the highest. It only sends you just over the 100AP mark with Nashor's Tooth, which buffs your W.
A lot of people have been saying that Guinsoo's isn't good on Kayle anymore. I disagree. Kayle still loves ability power, she still loves attack speed, and she still loves attack damage. The only thing Guinsoo's no longer provides her is its passive effect, but that passive is still valuable when you have Runaan's Hurricane. I've done a lot of testing and I strongly believe this is a powerhouse of an item on Kayle, it's just not an essential item anymore.
Note: Berserker's Greaves will put you over the attack speed cap with Nashor's Tooth, Runaan's Hurricane, and Guinsoo's Rageblade. You will lose about 24% bonus attack speed off of the rageblade passive. However, rageblade is usually going to be a last item, so how likely is it that you'll actually get there? I'd say if you want to go Berserker's Greaves then take them, and if the game goes on long enough to get rageblade then you probably have the luxury of just selling the greaves and getting different boots for full build.

Wit's End: The enemy team deals mostly magic damage and either they are stacking MR or your team also mostly deals magic damage (or both).
If you think you might need Wit's End, take Nashor's Tooth as your first item instead of Essence Reaver. Then, if you do need Wit's End, drop the rest of your build and just go on-hit Kayle. On-hit Kayle looks like Nashor's Tooth --> Wit's End --> Guinsoo's Rageblade --> Infinity Edge & Situational Item (last two in any order). Note: don't take Berserker's Greaves with on-hit Kayle--she will exceed the attack speed cap easily because of her passive, making them bad after level 11 and useless after level 16. If you need more damage, go Sorcerer's Shoes instead so your on-hit damage penetrates more.
If you didn't have the foresight to take Nashor's Tooth first, then don't worry about it and continue to build as normal, but skip Runaan's Hurricane because you'll exceed the attack speed cap too often for it to be worth the gold.

Guardian Angel: You are essential to the team, but you die more than you'd like. I would take this as my second-to-last item if it is needed, but it's also good as a last item if it gets to that point (Kayle isn't bad at the 'ole rush-the-open-nexus strat, since she has an invincibility ult on top of GA).

Mortal Reminder: You're playing against a fed Vladimir or something of that nature. Pick up this item as soon as it becomes relevant (usually 4th, 5th, or 6th item).

Phantom Dancer: This is comparable to Guinsoo's Rageblade in its usefulness. It puts out a little less damage, but it also provides you with movement speed that you could be lacking on the AD build, and a decent shield. A solid choice overall, and it maxes out your crit chance at exactly 100%. If you go the no-Nashor's build, this is a really solid replacement or 5th item. The drawbacks that the rageblade doesn't have are that this does not improve your magic damage very much, and it also doesn't double-proc your Runaan's Hurricane for the bonus multi-target damage.

Blade of the Ruined King: Your enemy has big health bars. Viable as 5th or 6th item in the Nashor's build, or as the replacement in the no-Nashor's build. Your lifegain wont be as high as with Hextech Gunblade, but it's enough to where you can bounce in and out of fights by using jungle monsters or minions to gain back some life.

Hextech Gunblade: This is a good item for sustain in the late game in general, but it's especially good if you find yourself in the kind of teamfights where everyone is just poking each other for a while before any real engage happens. Viable 5th or 6th item.

Mercurial Scimitar: This is for dealing with that one problematic ability that always gets you killed, such as Ahri's Charm.

The AP build has a little less single-target DPS than the AD build (not by a lot) and slightly higher multi-target DPS. Kayle can work with either build, and both builds are valid. I would say the AP build is just slightly harder to play, since screwing up an ability makes a bigger difference.

Taking the AP build you will have a slightly harder time before level 6, because on the AD build you can take Caulfield's Warhammer, which just offers more in both auto-attacking and poke than Stinger or Fiendish Codex do. Between levels 6 and 11, Kayle's poke will be about the same going AD or AP, but she'll have a slightly easier time in face-to-face confrontations on the AP build with Nashor's Tooth being the first item (it is very minimal of a difference).
Once both builds finish their second core damage item is when a playstyle difference will start to make an impact. On the AP path you have significantly more sustain with Hextech Gunblade. Your W is boosted, and you heal from damage dealt, so you wont have to back as often. Your poke will be a little more ahead of the AD path now, but the flipside is that your DPS will start to fall off, meaning you wont want to get in prolonged and consistent slapping matches. You'll definitely prefer poking and kiting.
By the time you've finished your third core damage item, the AD build path basically has you playing like a standard ADC, while the AP build path has some nuance.
After this, the AP build is very dependent on what situational items you take. Kayle's AP build can very easily transition into a hybrid build with AD situational items, and it wont ruin her damage output in the slightest.
Regardless of your build, by level 16 you'll probably just be able to do whatever you want within reason.

General guide to situational choices

This section is a work in progress.

This build path is designed around the need for Wit's End. If you don't need Wit's End, then you probably shouldn't be building on-hit (unless it's just for fun). It doesn't deal the highest damage and it doesn't have the best sustain, but it's the perfect kit for some niche situations.

On this path, your situational item is basically a choice of slow on magic damage vs slow on basic attack vs max health damage on-hit vs armor shred on-hit vs AoE damage on-hit. Keep in mind that taking Blade of the Ruined King on this path might push you past the attack speed cap when at full exalted stacks, depending on your runes, and Runaan's Hurricane definitely will.

This section is a work in progress.

Breaking down Kayle's damage output will allow us to decipher the best combinations for damage output. This is in a vacuum, meaning that only stats that affect Kayle's damage output are considered (e.g. we're ignoring things like movement speed and health). We are also ignoring invulnerability since we will rank up Kayle's ult when possible, regardless of how invulnerability factors into fighting. For our purposes, we only care about attack damage, ability power, attack speed, cooldown reduction, crit chance, and armor/magic resistance shredding. We will also count healing as DPS even though it is technically not. We can worry about stats like health, armor, and magic resistance later (we will probably assign arbitrary value to them, like "500 health is probably worth about 30 AP on Kayle"...that may not be accurate).

Unfortunately for me, we have to break down Kayle's damage output four times--once for each transformation. We will also break up her first two transformations into support and laning roles. This is a result of the assumption that, as support, she will probably deal most of her damage to the enemy via her abilities, whereas with laning we will consider basic attacks in the calculations. Her heal (which we are accounting for in our calculations) also has double the impact when playing support.

So you don't want to read the giant wall of text right below this? Okay. That's fair.

Here's a dump of practice tool results (burst data does not include her ult):

First Items (level 6)
Second Items (level 10)
Third Items (level 13)
Full Builds (level 18)
The DPS in this section includes using her Q and E as soon as they come up, averaged over 30 seconds.
Update:
I decided to redo the tests above using more realistic conditions.
After playtesting a lot I realized that I'm normally finishing my first item usually around level 9 at the earliest, not level 6. It can be closer to level 10 or 11 depending on what my first item is, if I had to buy buffer items, etc.

These are my new test results.
At level 9, assuming you have Mercury's Treads or Ninja Tabi:
Essence Reaver deals the most Q+E burst damage, as well as the most Q+E+R burst damage.
Guinsoo's Rageblade deals the most single-target auto-attack DPS.

At level 13, assuming you have Mercury's Treads or Ninja Tabi:
Essence Reaver + Guinsoo's Rageblade deals the most Q+E burst damage.
Nashor's Tooth + Hextech Gunblade deals the most Q+E+R burst damage.
Nashor's Tooth + Guinsoo's Rageblade deals the most single-target auto-attack DPS.
I was unable to test Essence Reaver + Blade of the Ruined King since the practice dummies have a ludicrous amount of health so the results were skewed.

My thoughts after this testing are that you should build according to what you're up against. For example, while I prefer the AD build, if you're going into lane against someone like Malphite, Sion, or Shen, who usually build some big armor item first (such as Sunfire Aegis, then the AD build might not be the way to go (depending on your playstyle; if you're looking to get kills then definitely don't go AD in that lane, but if you're looking to just farm safely, AD is still viable).

Let's start with levels 1-5. We'll assume that Kayle's damage output is 60% from abilities as a laner and 80% from abilities as a support for each of these calculations. This derives from game knowledge rather than statistics--Kayle will probably be bullied in this level range, so I doubt she'll be able to consistently engage an enemy. When I do more thorough testing I will update these ratios.

For Kayle's auto-attack damage let's assume she has 2 stacks of her passive. Again, this is because she probably wont be engaging often in these levels (but when she does, it's fair to say it would be for maybe 2-4 auto attacks on average). This means an extra 12% attack speed. Kayle's base attack speed is 0.695 + 0.07. At level 5 she has a bonus 6.2% attack speed (so 0.695 * 0.062 = 0.043 bonus attack speed). Accounting for the 2 passive stacks, giving 12% more (0.695 * 0.12 = 0.083), we get that Kayle makes 0.891 attacks per second.

Kayle's auto-attacks will not be exalted in these calculations, so we only care about armor. I'm using Zac as an example here, sine he has about average armor for a top laner. We can expect him to have 43.8 armor at this time, plus an additional maybe 10 from items and runes. Let's call it 54.

Fortunately the armor damage multiplier is easy math: damage = damage * 100 / (100 + armor). So Kayle's auto attack damage on our fake Zac will be AD * (100 / (100 + 54)).
Ah, but Kayle has an ability that shreds armor! And that armor shred doesn't scale with anything, so we can use it in our flat calculations!

Let's assume you can land Radiant Blast about 80% of the time. Since this is early game and you probably don't want to commit to fights, we will assume that the armor shred is active about 75% of the time you are auto attacking (assuming you use Starfire Spellblade to make a hit while you walk into melee range), making the armor shred active about 60% of the time (75% of 80% is 60%).
All together, Kayle's auto attacks deal AD * (100 / (100 + (armor - (0.60 * shred))).
Or AD * (100 / (100 + (54 - (0.60 * (54 * 0.20))))) = AD * 0.678 per basic attack.

Accounting for attack speed (we did some math earlier), Kayle deals (AD * 0.678) per basic attack at a rate of 0.891 + (0.695 * AS) attacks per second (at level 5, where AS is a percentage bonus attack speed expressed as a decimal number). Combine the two stats and we get that Kayle's non-ability damage amounts to (0.891 + (0.695 * AS)) * (AD * 0.678) damage per second.

Hmm, but something's missing. Do you know what it is? Crit chance! We simply take Kayle's previously determined damage per attack, multiply it by 2, multiply that by the crit chance (expressed as a decimal), and add that number onto the end of her DPS: (0.891 + (0.695 * AS)) * ((AD * 0.678) * (1 + (2 * CR))) damage per second.

And as stated previously, for levels 1-5 that will be 40% of her damage output in lane, or 20% as support.

Unfortunately, that's not the end of it, since Starfire Spellblade passively buffs auto attacks. Yikes. We'll come back to this when we figure out the ideal number of points to put in Starfire Spellblade by level 5, and then add its bonus damage (taking magic resistance into account) to our basic attack damage output.

Speaking of...
When accounting for abilities, we need to figure out a timespan over which we can calculate average ability damage. The radiant blast cooldown at level 1 is 12, whereas the other two abilities have constant cooldown times (we obviously ignore her ult for now). We'll have to take Kayle's CDR into account for that Q, but otherwise we have easy calculations. The common denominator of 12, 15, and 8 (her ability cooldowns) is 120, so we'll go off that (that means we'll assume 10 Qs, 8 Ws, and 15 Es at lvl 1 of each).

Each ability's on-cast damage over 120 seconds can be calculated as (number of casts in 120 seconds) * (damage on cast).

Radiant Blast deals (120 / ((13 - pts) * (1 - CDR))) * ((70 + (pts * 45)) + (0.6 * AD) + (0.5 * AP)) magic damage over 120 second (pts is the number of points placed in Q). Taking into account MR (following similar logic as with Zac, but for Galio, we'll say the enemy has 35.9 + 8 = 44 MR) we get true damage = dmg * (100 / (100 + MR)) (0.695 in this case), where dmg is the previous calculation of damage.
(120 / ((13 - pts) * (1 - CDR))) * ((70 + (pts * 45)) + (0.6 * AD) + (0.5 * AP)) * 0.695. Divide by 120 to get damage per second.

Celestial Blessing offers (120 / (15 * (1 - CDR))) * ((60 + (pts * 30)) + (0.3 * AP)) true healing per 120 seconds. Multiply this value by 2 for the support role, since both champions get equal healing. Divide this value by 120 to get damage per second.

Starfire Spellblade deals (120 / (8 * (1 - CDR))) * (7.5 + (pts * 2.5))% target's missing health. For health we'll assume 50% missing health and use Nunu as our example, making the average max health of our enemy (848 + 150 = 998), and therefore the damage is now (120 / (8 * (1 - CDR))) * ((7.5 + (pts * 2.5)) / 100) * (998 / 2). All of that put through magic resistance making the true damage = dmg * (100 / (100 + MR)). Except this time, we can say that there is a chance the enemy's magic resistance has been shredded by 20%. Their MR will be shredded on average (120 / ((12 - Qpts) * (1 - CDR))) * 3 seconds. But we probably wont be spamming abilities because of mana constraints so let's simplify it for ourselves by saying their resistances will be shredded about 12 seconds out of the 120 (10% of the time). This gives us the calculation of true damage = dmg * (100 / (100 + (44 - (0.10 * (44 * 0.20))))) (comes out to 0.725).
(120 / (8 * (1 - CDR))) * ((7.5 + (pts * 2.5)) / 100) * (998 / 2) * 0.725. Divide this by 120 to get damage per second.

But wait, that's not all! Starfire Spellblade also has that passive effect that adds damage to basic attacks. Since we are operating under the assumption that Kayle's basic attacks only deal 40% of her laning damage (for the purposes of stat optimization we're counting this as a basic attack damage, but for purposes of skill selection we're counting it as ability damage), we will multiply all expected damage from this ability's passive by 0.40. That means that it deals ((5 * pts) + 5 + (0.10 * AD) + (0.20 * AP)) damage per basic attack, which is at a rate of (0.891 + (0.695 * AS)) attacks per second. Since our ability damage is over 120 seconds, and 40% of 120 is 48, we get that this ability's passive damage (which we'll add on its active damage) is 48 * ((5 * pts) + 5 + (0.10 * AD) + (0.20 * AP)) * (0.891 + (0.695 * AS)). Add in MR with shredding at a rate of about 75% of auto attacks (as we did for auto attack damage), and we'll get dmg = dmg * (100 / (100 + (44 - (0.70 * (44 * 0.20))))) for a grand total of ((5 * pts) + 5 + (0.10 * AD) + (0.20 * AP)) * (0.891 + (0.695 * AS)) * 0.725 * 48 damage over 120 seconds (again, divide by 120 to get damage per second).

Whew.

Unfortunately we are at a fork in the road. There are two approaches we can take from here. Either figure out which abilities to max first and then determine what stats to prioritize from that, or figure out which stats to prioritize and then work out which abilities to max from there. While we could break down each ability to determine how much damage it gains from each stat (some will be constant damage gain, and others might be damage multipliers) and then use that to determine where our stats should be (and this result would vary as the game goes on since Kayle will eventually have all her skills maxed), I actually like the former approach because it allows for some wiggle room to use game sense...and also it's easier...

Let's take a break then, and figure out how to max the abilities levels 1-5. There is one thing to keep in mind: levels 1-5 Kayle does not have a passive cleave that benefits from Starfire Spellblade (the attack speed still does, though), whereas at level 11 Kayle becomes fully ranged (making reaching exalted easier) and gains more bonus attack speed, making Starfire Spellblade extra impactful. We'll ignore this on support, but for laning that means we definitely want a maxed out E by level 11 (actually level 10 since we will put a point in her ult at 11). In the early game, though, we probably wont expect the opponent to be at low health very often, at least not until ganking season.
Basically, this means we probably take E, Q, W, Q, E, R, E, E, E, Q, R, Q, Q, W, W, R, W, W as laning Kayle. I think taking E or Q first is interchangeable; E is going to help you not screw up those early minion kills, whereas Q is better for poking since some of E's damage is based on missing health and at level 1 no one is missing health.
As support, naturally we'll want to take W as long as we're in true laning phase. Her support skill progression probably looks like E, W, Q, W, W, R, Q, E, E, E, R, E, Q, Q, Q, R, W, W, with the first points in W and Q being interchangeable depending on how the lane is going. We don't take W after level 6 as support simply because it becomes less useful as laning phase comes to an end (and you're not planning to play support anyways, right?--you're just pretending to play support until you're carrying).

Okay so we can fill in the pts variable now, assuming Kayle is level 5 and laning.

Auto-Attack Damage: 40% (0.4x multiplier)
Standard Attacks: (0.891 + (0.695 * AS)) * ((AD * 0.678) * (1 + (2 * CR))) damage per second.
Starfire Spellblade passive: (15 + (0.10 * AD) + (0.20 * AP)) * (0.891 + (0.695 * AS)) * 0.725 damage per second.

Ability Damage: 60% (0.6x multiplier)
Radiant Blast: (120 / (11 * (1 - CDR))) * (115 + (0.6 * AD) + (0.5 * AP)) * 0.695 / 120 damage per second
Celestial Blessing: (120 / (15 * (1 - CDR))) * (90 + (0.3 * AP)) / 120 healing per second as a laner, or (120 / (15 * (1 - CDR))) * (150 + (0.3 * AP)) * 2 / 120 healing per second as a support.
Starfire Spellblade: (120 / (8 * (1 - CDR))) * (12.5 / 100) * (998 / 2) * 0.725 / 120 damage per second.

So, uhh...What items are the best for Kayle levels 1-5 then?
Well, we're just going to evaluate each variable independently one-by-one, substituting the other variables with their level 5 base value, and look at the damage curve. That isn't completely accurate though, since some stats scale with each other.
For example, on most mages, if you were to build AD instead of AP, then there would be no point in getting CDR because none of your abilities would deal a ton of damage, and the cost-performance ratio of CDR is lowered as a result. On the other hand, attack speed would benefit more. Attack speed is basically the CDR of Crit Chance and AD. Also as a friendly reminder, Crit Chance is designed to equate to 1% bonus damage per 1% Crit Chance. Therefore, Crit Chance usually only starts becoming "worth it" once the cost of buying 1% of your AD as AD is higher than buying 1% of Crit Chance. For most champions, that is at 115 AD. However, Kayle has a passive cleave that can critically strike, making crit more valuable at lower ADs than 115. However again, Kayle's E cannot critically strike. It's all very complicated stuff and I hate it.
Back on topic: Graphs will show you how each stat affects Kayle independently, but not in conjunction with each other. I don't know enough math to do that by hand, so I'm going to brute force those results by running them through a simulator.
How will that simulator work? I'm going assign each stat a gold value (see https://leagueoflegends.fandom.com/wiki/Gold_efficiency), and assign Kayle an amount of gold she can use (what amount? let's just call it 2,000, but that guess can be improved by just playing games and taking the average gold earned pre-lvl 6). I will then tell Kayle that she can spend gold on stats in increments of 100 gold, and she will then spend that gold in every possible combination of ways on stats (she will first spend it all on damage, then she will spend 1900 on damage and 100 on attack speed, etc.). The simulator will calculate her damage output in each of those scenarios, and it will report back which scenario had her dealing the most damage.

The exact code I'm using for the simulation is:
https://pastebin.com/FZBv9M7Q

Here is a graph of each stat's impact on damage per second independently (AD is dark blue, Attack Speed is green, AP is purple, Crit Chance is black, CDR is light blue). Note that Attack Speed and Crit Chance are going by percentage, not actual value. Also note that this is the impact of bonus stats of each type, this does not include the impact of base stats (because you can't change them).
My simulator gave me this as the optimal combination of stats:

So as you can see, the simulator strongly favors going the AD route. That makes sense given she doesn't have her cleave yet, she doesn't have her ult yet, and we aren't factoring in the advantage of armor/mr shredding in a team fight. Nevertheless, I highly doubt the simulation is going to spontaneously start favoring AP as Kayle levels up. My gut feeling at this point is that she's going to be best suited as a late-game AD carry

But hold up. Don't get ahead of yourself. Just because the simulator said so doesn't make it so. Unfortunately, you can't just pay gold for stats. There are many items and they all have unique attributes. There's no way in hell I'm going to do all this math for each item individually (hence why I'm doing it by stat instead of by item) (EDIT: I ended up doing this for each item individually, just not including the passive effects on items, so it's not really accurate). This is just a guide. Not to mention that you should probably start building towards an item you'll actually need late game, rather than what helps you most early game. Honestly this level 1-5 section is mostly just a warmup. What we'll really do is figure out what items she wants most levels 6-10, and then come back here and decide what possible purchases we could make levels 1-5 that both align with her desired stat distribution and help build towards one of her level 6-10 items.

So that's it for this chapter. We'll circle back after our levels 6-10 analysis and decide what items she should buy in this phase.

Levels 6-10 add more complexity because of the passive cleave (which can critically strike). Kayle also gets her ult here, and since we're using 120 second frames for our calculations, we'll start to subtract damage output from basic attacks accordingly, since she can't attack while ulting.

For this section we will assume that Kayle can reliably get about 3 exalted stacks. We'll use that number for the attack speed bonus we use, and we'll also use that to say that 3/6 or 50% of the time she will have her cleave active.

We will also say now that Kayle is dishing out auto attacks 55% of the time she's hurting someone and 45% of the time it's ability damage (again, that does not mean 55% of her damage output is from auto attacks and 45% is from ability damage).

Her auto attack damage calculations can stay the same after we update the pts put into E, with the one addition being her cleave will be assumed active 50% of the time she's auto attacking.
We start by updating her base attack speed at this point to be 0.695 + (0.695 * 0.176) + (0.695 * (0.06 * 3)) + 0.07 = 1.01242 (up from 0.891).
Her opponent's armor is going to be assumed 30 higher now (84), and since we can be a little more aggressive now we'll say that the 20% armor shred is only active about 45% of the time she's attacking, making the armor multiplier (100 / (100 + (84 - (0.45 * (84 * 0.20))))) = 0.569476
I'm also going to make an amendment to my previous equation, because it used Kayle's total AD as the multiplier. I'm changing it to be based on bonus AD so it's more obvious what to do when we run the simulation.

Auto-Attack Damage: 55% (0.55x multiplier)
Standard Attacks: (1.01242 + (0.695 * AS)) * ((AD * 0.678) * (1 + (2 * CR))) damage per second.
Starfire Spellblade passive: (15 + (0.10 * AD) + (0.20 * AP)) * (0.891 + (0.695 * AS)) * 0.725 damage per second.


This section is a work in progress.

At levels 11-15, Kayle is fully ranged, and she gains more bonus attack speed. Because of this (calculations aside), I value Starfire Spellblade the most at this level, as it's significantly easier to reach exalted. I mentioned this before. We will assume that Starfire Spellblade is maxed out.

This section is a work in progress.

Our calculations for levels 16-18 are the same as the calculations for levels 11-15, except that we don't account for opponent armor and magic resist when evaluating Kayle's passive. We also have the luxury of assuming now that all of Kayle's abilities are maxed out.

This section is a work in progress.