Hello, my name is forfor. I'm a gold-ranked Zilean support main who owns every champ, and has been playing since season 2. (Do you remember AP assassin Tristana? Pepperidge Farm remembers)
So, if I'm a Zilean main, why am I writing a guide for Irelia? Because I have ADHD and a love for off-meta builds. Despite my attention deficiency, I read the patch notes religiously. Whenever champs experience mechanical changes, I love to theorize, analyze, and test to seek loopholes, and exploits.

Because I'm lazy, and don't know much about html. Sorry :(

TLDR; you can burst for 475% ap ratio with only e-w-r, 2 q procs and everfrost. about a 4 or 5 second combo.

While reading the recent patch notes, I noticed something: her W now has 120% ap scaling at max charge. This got me thinking, and I did the math on all her abilities. if she lands e, w, and the R projectile, her base AP ratio before items and runes is a whopping 270%. It becomes 340% if they then walk through the sword field. This number would be more than acceptable on plenty of ap assassins. For reference, Leblanc has 280% ap scaling if she hits every single one of her abilities including the second proc on her chain. I did more math. When you count runes and my recommended first 3 items, she can do a whopping 475% AP scaling on a single burst. Admittedly, this number carries several assumptions:
-The enemy walks through the blades after ult at least once. Every time an enemy passes through the blades, the damage proc has a 70% ap ratio. If you don't want to factor that assumption into the end result, then the value is 405%
-I only assumed two procs of q, and one proc of lich bane. This is really the absolute minimum possible number of on-hit effects for a normal fight. The end number varies wildly depending on how many times you use q, and how many times you auto. You can easily deal over 5 or even 600% ap scaling via nashors and lich bane if the enemy gives you time to do more q procs and autos.
-I didn't do the math on base damage values compared to a normal assassins base damage values. Irelia's base damage on w and e are decent, and the W has insane scaling, so there shouldn't be a real issue here.
-I didn't factor in the bonus damage from max stack passive. This adds as much or almost as much damage as nashors until you're 2-3 items into your build, and shouldn't be taken lightly.

If you want to read the exact, detailed calculation I did, I posted it here:
https://www.reddit.com/r/IreliaMains/comments/ogu4ob/does_anyone_else_think_ap_irelia_seems_viable_now/

I'm not going to lie, Ap Irelia is squishy. AP items simply don't offer the same level of beefiness and sustain that you can get from AD bruiser items. Anyone who walks into this build and tries to act like a normal Irelia is probably going to die. A lot. Instead, remember that AP Irelia is functionally an assassin. Choose a priority target, burst them, peace out, and reassess. (or burst someone else who got hit by your ult) If you don't like that playstyle, that's fine. This build may not be for you.

Passive: Ionian Fervor
Irelia gains a stack of Ionian Fervor for every enemy champion hit by her abilities, or if she hits at least one non-champion, stacking up to 4 times. Ionian Fervor lasts for 6 seconds, with basic attacks against enemy champions and large monsters and ability hits refreshing the duration.

Ionian Fervor: Irelia gains 7.5−20% bonus attack speed per stack, up to 30−80% at maximum stacks (depending on level).

At maximum stacks, Irelia's basic attacks deal 10−61 (+0.3% bonus attack damage) bonus magic damage on-hit (depending on level).

Enemy champions and large monsters hit by Flawless Duet and Vanguard's Edge's initial barrage are marked for 5 seconds. The mark can be consumed by Bladesurge.

This ability offers much more power than you likely expect it to just from reading it. The attack speed and on-hit damage are game-changers in many fights. Victory or defeat is likely to be determined by how efficiently you stack your passive in fights. (or before fights if you can manage it) Note that as an AP build, we don't benefit from the 30% bonus AD scaling so the base number is what we get. Even so that's still almost as much as your nashor's on-hit damage for at least the first half of the game.

Q: Bladesurge
Irelia dashes 100 units through the target enemy's location, and upon collision or dash completion, she deals 5 / 25 / 45 / 65 / 85 (+0.6% attack damage) physical damage, applies on-hit effects, and heals herself. Bladesurge deals 55−259 bonus damage to minions (depending on level).

Bladesurge's cooldown is reduced to 0.2 seconds upon consuming the mark from a completed dash, and resets if the target dies to or during Bladesurge.

The bread and butter of an AD build. Less so here, but still very important. The ability to apply several hits of nashor's, lich bane, and passive on-hit damage in quick succession with no auto attack reset timer is extremely powerful. This is your primary mobility tool, your primary farming tool, and even offers a moderate amount of sustain.

W: Defiant Dance
Irelia charges for up to 1.5 seconds, during which she increases Defiant Dance's damage over the first 0.75 seconds of the channel and reduces incoming physical damage by 40−80% (+7% per 100 ability power) (depending on level) and incoming magic damage by 20−40% (+3.5% per 100 ability power) (depending on level). Defiant Dance's charge cannot be interrupted by crowd control.

Defiant Dance can be recast within its duration, and does so automatically after the duration.

RECAST: Irelia swipes with her blades in the target direction, retaining Defiant Dance's damage reduction for 0.5 seconds after the recast is initiated and dealing 10 / 25 / 40 / 55 / 70 (+0.5% attack damage) (+40% of ability power) physical damage to enemies around her and within a line, increased by 0−200% (depending on channel time) for a total of 30 / 75 / 120 / 165 / 210 (+1.5% attack damage) (+120% of ability power) physical damage.

Defiant Dance's recast can be used while affected by cast-inhibiting crowd control. If Ionian Fervor would expire during the charge, it will instead refresh every 0.25 seconds for the remaining charge and be retained for 0.5 seconds after the recast is initiated.

This ability requires careful timing, but can chunk an enemy for a monstrous amount of damage on a single hit when charged to a high value. Not only that, but the damage reduction and cc nullification are very powerful for winning trades, countering assassins, or just buying a couple extra seconds while your team kills someone that's targeting you. It's worth noting that because the damage reduction scales with AP, you'll reach 100% physical damage reduction by late game. This ability also adds huge waveclear. Once you have at least a blasting wand, a single fully charged cast will kill caster minions from full HP, and likely bring melee minions low enough to kill with Q. Use this to hard-shove waves, save towers, and split push quickly.

E: Flawless Duet
Irelia sends a blade to the target location for 3.5 seconds. Flawless Duet can be recast after 0.15 seconds while the blade is active, and does so automatically at the end of its duration, though not if she is channeling Teleport.

RECAST: Irelia sends a second blade to the target location, or to her current position if Flawless Duet was recast automatically.

Once both blades have been placed, they fly toward each other regardless of distance and converge over 0.25 seconds, afterwards dealing 80 / 125 / 170 / 215 / 260 (+80% of ability power) magic damage to all enemies within a line between them and stunning them for 0.75 seconds, during which they are revealed.

Flawless Duet's recast can be used during Bladesurge and the cast time of Vanguard's Edge. Each of the blade's travel times are 0.264 seconds.

Your primary tool for initiating, trading, and escaping as AP Irelia. It has high base damage, decent AP scaling, and allows for q resets on champion and large monsters. The cooldown starts out high, but reaches a manageable 12 seconds at max level. (further reduced by ability haste) The second dagger can be tricky to land, and its best to wait out your enemies reaction for a second before you try to land it. The list of tips and tricks for baiting an enemy into eating your second dagger is too long for this section.

R: Vanguards Edge
PASSIVE: Reduces the cooldown of Bladesurge to 0.5 / 1.5 / 2.5, applying before ability haste.

Irelia launches a barrage of blades in the target direction, expanding outward upon hitting an enemy champion, dealing 125 / 250 / 375 (+70% of ability power) magic damage to all enemies hit and briefly revealing them.

After expanding, the blades drop on the ground, knocking all enemy units away from them, though not rendering them airborne, and forming a spade-shaped perimeter around the enemy champion hit for 2.5 seconds. Enemies that pass through the blades for the first time are dealt 125 / 250 / 375 (+70% of ability power) magic damage and are slowed by 90% for 1.5 seconds.

The perimeter will point at the same angle Irelia casted it from, but based on the target's center.

The heart of your power. Most fights you enter into will revolve around "Do I have my ult or do I not have my ult?" Remember that every time this ability does damage, it adds 70% AP to the damage. That means that on the initial hit, you do 70% scaling to everyone in the aoe, then if they run through the swords, it does another 70% scaling. Every hit also stacks your passive. Every hit also adds a q reset proc. The aoe for this effect is fairly large. So what happens when you land this on 3-5 enemies who are all grouped close together? Absolute carnage. Just remember that if you jump an enemy too early, they're more likely to turn and fight you rather than running through the swords.

Nashor's Tooth: The heart of your build, and the absolute number one must buy. Everything else is optional. My mythic and third item are preferences, but can be traded out to experiment with other things. This item cannot. The number one weakness of AP Irelia is that her Q lacks any AP scaling. What it does have is the ability to apply on-hit affects. That means nashors will add 20% ap scaling to every q application. Combined with the natural damage from her passive, she can do an insane amount of damage very quickly. Not only that, but the attack speed it provides is a vital stat for most situation. Take this as your first item, no matter what.

Everfrost:
AP Irelia isn't like AD Irelia. She doesn't act as a drain tank hyper-carry. AP Irelia is a melee assassin, and should be built as such. So why everfrost? A few reasons. First, the active is broken. Every other mage mythic passive/active has 10-15% ap scaling. For whatever reason, Everfrost has 30% AP scaling. So it does a bit more damage than other similar effects. Not only that, but the cc is amazing. Need to escape? Use Everfrost. Want help landing E? Use Everfrost. There are very few situations where "use Everfrost" is a wrong answer. On top of all that, the stat lineup is perfect. The HP helps Irelia's survivability, the AP is comparable to the other mythics, the mana is enough to ensure you'll never run out, and 20% ability haste is the icing on the cake. The legendary effect also stacks up to nearly a full extra AP item if you reach full build. (more realistically, you'll get 15-30 over the course of a game which is still pretty good. Equivalent to an extra 600ish gold)

Lich Bane:
Did I mention that Q applies on-hit effects? how would you like to add an extra 40-80% AP ratio to your bursts? (plus 60-75 extra base damage per spellblade proc) Getting more movespeed is also nice for a champion as mobility-reliant as Irelia. I'm honestly 50-50 on whether to get this item second or third since it adds so much more damage than everfrost, but has less utility and general stats. The answer is most likely situational, so it's up to you.

I usually start with a dorans blade and a pot against most champs. That being said, there are certain utterly obnoxious champs like xerath or lux who can just sit far beyond your engage range while lobbing spells at you. Against these types of lanes you want a dorans shield.

Early game you want to focus on stacking and maintaining your passive if you can do it safely. Once you have it stacked, you should look for an opportunity to trade. Don't over-commit to any one trade. An important feature of Irelia is that her q offers a lot of sustain, so in most situations an even trade is a victory for you since you can quickly regain a big chunk of any HP you lost.

An easy level 1 trade is to stack 3 stacks against the first 3 melee minions. In most situations, they should lose HP at almost exactly the same rate. Quickly dash between them so that you kill all 3 near-simultaneously, then perform another dash onto your lane opponent. If you did it right, you should get 3 stacks from killing the minions, and your 4th stack from the dash to enemy. Land two autos, proc electrocute, and walk away. This will be an even trade against most enemies, but your sustain usually makes it a win for you. don't do this if they out-sustain you, or have strong level 1 cc.

If you can't safely stack your passive, don't even try to fight. Let them shove you to tower, and farm as best you can under tower. Irelia is very strong against tower-dives due to the damage reduction on her W, and has a very easy time farming under tower due to Q.

Always be looking for opportunities to gank other lanes. Irelia is very strong if she gets ahead, and in general supporting other lanes puts your whole team in a better position. Really this is advice that applies to any mid-laner. Just watch out on your way back because a lot of people will try to ambush you when you return to lane.

As a mid-laner, you are partially responsible for tracking and setting up dragons, and dragon pit fights. This is just general advice, not Irelia specific. What is Irelia specific is that if you have ult, you thrive in these kinds of large fights. Try to set up multi-person ults, then assassinate squishies as they run away through the swords. Even if all you accomplish is chunking a few people for decent damage, and/or getting a flash, you've still set your team up to be in a more advantageous position when it comes to securing dragon.

If you're still learning Irelia, make sure you play passively until level 3. You at least want to have your full toolkit before you think about committing to a real fight.

I'm defining mid game here as "whenever you finish nashors." If you finish it early, around the ten to fourteen minute mark, then you're in a good position. Feel free to be aggressive in fights, and look even harder for chances to catch out enemy champs. As long as you continue to play smart and land your abilities, you can and will 100-0 most enemies.

If you finish it late, or fed your lane? Don't panic! Between ult, e, and eventually everfrost, Irelia has quite a bit of cc. Transition to peeling for a more fed ally, or zoning enemy teams with ult in order to make them easier for your allies to kill. Either way, play passively and watch for quick, easy kills on enemies that are low or out of position.

Irelia is also very good at split pushing, so if you need to catch up on farm, feel like you aren't contributing much, or want to press the advantage for your team, consider dipping into a side lane for a while. Just keep in mind that Irelia's escape tools aren't the best, so if you want to do this, do it safely. Be very conscientious about warding, and watch the map like a hawk so that you catch it if the enemy team mysteriously vanishes.

Sadly, games rarely make it to late game these days. If they do though, remember to play it safe. Damage numbers have likely gotten fairly high, while your durability probably hasn't increased much. A single mistake can cost you the game at this point. The solution? Don't be in a place where you even have the opportunity to make a mistake. Play cautiously, let your team draw aggro, and look for a chance to pick off enemies while they're distracted by your allies.