Flash is generally a core spell, there's no reason to switch it out for anything. Exhaust tends to be a stronger pick throughout the game, as it gives you a massive damage debuff that is particularly effective against high damage assassins like Zed, but picking up Heal while your carry picks up Barrier can also be a strong option. Ignite can be picked up against a high sustain team or against a high sustain support like Soraka, if necessary.
I'm a support player from the North American server. I play primarily Karma, Taric, and Lulu. I've always loved Kayle as a champion and I've been experimenting with her as a support for quite some time.
Kayle is definitely not a common support pick. She saw a minuscule amount of play, by professional players, as a counterpick support earlier on in the season. At this point she is hardly picked at all and I'd say she is a pretty good example of a functional product. She works as intended (if played properly), but she is nothing special and offers very little over other support picks. This particular guide and its included builds focus somewhat on Divine Blessing. I really cannot recommend this as a staple pick, especially if you are inexperienced with Kayle, but it can work.
Kayle has elements that can make her a playable support with practice. Her sustain is surprisingly decent, she can kite fairly well, she can help shove up a lane. The ability to switch between melee and ranged allows her to fully utilize items that other supports of a similar type cannot. She can also a force to be reckoned with in teamfights if you know how to time Intervention properly, especially in conjunction with Divine Blessing.
Kayle requires extremely perceptive play and very good timing to play successfully as a support. A strong sense of when to use Intervention is extremely important for this to work at all. Without that, you might as well play another support entirely. Divine Blessing has a very high base value, but it has a long flat cooldown to compensate, which can render certain match-ups nearly unplayable. Kayle can also have resource problems if she is forced to spam Divine Blessing (though it is actually pretty economical for its mana cost), but this can be compensated for with rune choices fairly easily.
Holy Fervor: Reasonably helpful for raising allies' damage in trades. One stack is applied by Reckoning by default, which is somewhat helpful in applying it.
Reckoning: This skill is has reasonably high base damage and is good for kiting single targets. The slow scales up with ranks, starting with an already respectable 35% slow. It also applies a stack of Holy Fervor, lowering the target's armor and MR by 3%. The latter part is useful when followed up by Righteous Fury in order to further shred their armor and MR over the course of trades.
Divine Blessing: This has a surprisingly high base, which, when further enhanced by Revitalize and the like, can be pretty good for a strong burst of healing. Its mana cost is pretty forgiving for the amount of health it returns, but this is largely balanced out by its flat 15 second cooldown.
Righteous Fury: This allows for some interesting interactions. You can use items like Relic Shield and Knight's Vow to their full potential as long as this is not activated. However, activating this for the ranged attacks can be very useful when clearing wards, attacking turrets, trading, or if you want to help shove the lane up.
Intervention: This is pretty much the bread and butter of support Kayle. Your ability to properly time this skill is the most important thing you can practice if you wish to be successful. Using this properly can completely turn around trades, win team fights, or just save an ally from an otherwise untimely demise.
Currently, maxing Divine Blessing first is the best option on support Kayle as it gives her strong sustain. I usually prefer to max Reckoning second, as its slow scales up with ranks and the base damage is fairly high on its own. I then max Righteous Fury last, as you really only need one point to get the benefit of having it. Intervention should be taken whenever it's possible to take it.
Starting Items Any of these are fine starting items for Kayle. She can use all of them pretty well. Ancient Coin offers the best gold generation and sustain, so I tend to prefer it in most situations. Spellthief's Edge is good if you plan to be very aggressive, but it's lackluster in any other context. Relic Shield is a nice alternative to Ancient Coin if you want to be a bit tankier to start, but you'll noticed a marked difference in your mana sustain.
Core Items As some type of warding item is core on supports in general, mentioning it seems almost redundant. However, the extra health that Eye of the Aspect provides is quite beneficial, while Eye of the Watchers gives you a decent chunk of AP. Eye of Ascension gives a nice gold/mana passive that requires little to no effort on your part to use. Oracle Lens is another core support item, that should be bought as soon as you finish the gold item quest and get the warding passive, as it allows you to gain vision control easier, which is important for keeping an eye on objectives such as Baron Nashor and Dragon.
Defensive Items These are the best defensive purchases for Kayle currently. Redemption is just an all around good purchase on most supports and Kayle is no exception. It increases her healing and gives her a nifty active that can be really useful if used properly. Mikael's Crucible is very strong against compositions with a lot of hard CC, it can save an ally from an untimely death if the active is timed properly. Locket of the Iron Solari should be purchased against teams with a lot of Magic damage, especially if it's AoE, but is strong against any high burst team compositions due to the shield active. Banner of Command is sort of a budget Zz'Rot Portal and, while cheaper, is more situational as the promoted minion is only immune to magic damage.
Offensive Items Kayle largely doesn't need to purchase offensive items. However, there are a few items that can be beneficial to purchase on her. Ardent Censer gives her a good chunk of AP, CDR, and movement speed. However, its primary appeal is the passive it applies to your healing, which gives targets healed or shielded attack speed and on hit magic damage. This makes it useful to rush with ADCs who utilize attack speed well, especially if they would normally get Runaan's Hurricane as one of their core items. It also further increases her healing values, which is great later on in the game. Zeke's Convergence is a strong purchase if you wish to further buff an ally's damage. Intervention has a fairly low cooldown and you can easily "ignite" the frost storm with Righteous Fury. Athene's Unholy Grail can be a decent purchase if you're ahead and just want to take your healing to silly levels. The blood charge passive the item provides is fairly easy to stack due to Reckoning's high base damage, plus the extra AP scales nicely into both Reckoning and Divine Blessing.
Situational Purchases These items are situational, but can be useful in or against certain team compositions. Zz'Rot Portal is a pretty decent purchase if your team is getting pushed in or if you want to start an AFK push in one of the lanes to force the enemy team to send someone to deal with it. Frozen Heart can be a good purchase against attack speed dependent champions or against those who utilize attack speed as part of their damage ( Yasuo, Lucian). However, the lack of health in the item itself means you need to itemize health to make the best use of its armor. Knight's Vow can be a nice purchase if you want to offer a little more protection to an ally. Remember that its effects are reduced when Righteous Fury is active, so if you're using that frequently, I'd hold off on this purchase. Twin Shadows is a fairly good option against stealth champions, as the ghosts will chase champions who enter stealth after it's activated. It's also a good scouting tool, if needed. Ninja Tabi are a strong purchase against high AD teams with champions that rely primarily on auto attacks.
Largely Kayle's function in lane is to keep her ally sustained with Divine Blessing, while using her other abilities to apply Holy Fervor to help her carry get an edge in trades. The early part of this lane can be hard to play properly, as Divine Blessing does not have a particularly forgiving cooldown. Intervention allows her and her carry to pursue riskier trades, but care should be taken as to not get too confident and throw advantage out the window pointlessly. Largely, timing on abilities like Intervention and Divine Blessing will be key.
The focus here is basically the same as before, only now the stakes are higher. You'll have to pick and choose which allies to use Intervention on. Occasionally this can be yourself, but normally shouldn't be. You can also bait out key abilities from the enemy team and use Intervention on yourself to wait it out while your allies move in, but I don't really recommend this most of the time. Knowing when to do what with Intervention is a really important part of playing Kayle successfully as a support. The cooldown gets really low later in the game, at which point this becomes slightly less important. Divine Blessing should primarily be used to give a burst of healing, though it can also be used to give a movement speed boost if the healing itself isn't necessary. Reckoning can be used to kite enemies or to slow down primary targets, whichever is appropriate. Righteous Fury should be used to apply more stacks of Holy Fervor
Well, here's an example of a pretty clutch Intervention by yours truly:
You'll find that timing is a huge factor in your success. It's worth it to go into the practice tool and purely practice casting Intervention if you want to do this seriously and you're not that experienced with Kayle. With practice, you, too, can be the talk of the town with your last second death prevention. There's not a lot else to be said about it, though it does help if you follow it up with Divine Blessing too.
Kayle mostly works well with champions who can deal a decent amount of damage on their own and don't need their support to bring a lot to the lane. It also helps if they have strong ultimate abilities or their own hard CC, as Kayle's only CC is a single target slow. In general, Kayle works well with champions who have strong trading potential, but might need protection due to innate fragility/low mobility.
Divine Blessing has a flat 15 second cooldown, which can make dealing with champions like Zyra and Karma really difficult. Caitlyn can easily stay far enough away to make trading difficult before six, which can make her an annoying champion to lane against. Generally, Kayle tends to work best against low damage supports or tanks that can be easily kited. Malzahar is somewhat of an exception to this rule; a properly timed Intervention will mostly or completely nullify Nether Grasp.