You listed things in the cheat sheet that are not explained in the main guide:

• Overall: Stat shards
• TOP | AP Build: Absolute Focus, Quicksilver Sash (explained at AD builds, but isn’t featured in the AP build)
• MID | AP Build: Demolish, Quicksilver Sash
• Top & Mid | AD Build: Sanguine Blade, Stormrazor
• TOP | Muramana Build: Absolute Focus, Lich Bane (featured in AP build, but isn’t featured in Muramana build)
• SUPPORT | Standard Build: Arcane Comet, Knight's Vow
• SUPPORT | Demi-Carry: This entire build isn’t mentioned at all!

Besides that there isn’t much to say about the cheat sheet, except that there aren't any notes. Maybe you could add some to the different cores/items. Yes, the main builds (AD/AP/Muramana) would still go unexplained, but it might make things a bit more clear.
You mention Elixir of Skill here. I am quite sure this item isn't available anymore after the removal of Kleptomancy.

At Celestial Blessing: “if you spam this without mana items or having some form of mana regen”. Although this sentence is (probably) grammatically correct it is a bit confusing to read, perhaps adding another “without” before “having” might help.

At the skill order you say you should go the 3Q > E max when you cannot effectively trade back until you are level 6. I have no clue what you mean by this (you can just smack them with Q ?). Is this matchup specific? Rune specific? Or perhaps it is the standard ability order? Now, what usually happens with these kind of champs/runes sections is that there is this mountain of random unusable information (ever heard of the choice paradox?). You managed to combat this pretty well and got a fresh overview of possible runes. However, the formatting came at the cost of some information & it isn’t consistent:

• Press the Attack page: Legend: Tenacity not listed in the table at alternative choices & Celerity is not explained although it is listed as an alternative choice
• At Fleet Footwork, the Bone Plating code in the table is broken (shouldn’t it be an alternative choice to Second Wind btw?). You also don’t say anything about Demolish anywhere.

The section explains really well why certain runes work, but it lacks a distinction in when certain rune paths are good/should be used. Right now it feels like Press the Attack should be run with Sorcery & Lethal Tempo should be run with Domination and not with other secondary trees. The secondaries might be very matchup specific, but even adding a generic explanation on which keystones have synergies with which secondary trees or even mentioning what a particular secondary tree can offer will help.

I am also not sure whether all runes are as viable as you say. My main concerns here are Demolish and Approach Velocity. I feel like Kayle never really ends up hitting turrets (especially not in the laning phase) and she doesn’t need the extra ms from Approach Velocity because Celestial Blessing gives a ridiculous amount already.
Again, a mountain of information becomes readable with clean formatting. Here are some notes:

• It is not clear whether there are two possible cores on AP toplane itemization or whether its just 1 with the other being *nice to know*. If it is the former, then it is not clear when one core should be taken over the other. If its just 1 being viable: why bother mention the other option at all? The same happens at the cores of the AD & the Muramana builds.
• Perhaps you should explain why Runaan's Hurricane is in an.. AP build? Also, I don’t know why you would ever mention Phantom Dancer here as well.
• You mention Luden's Tempest very briefly in Lich Bane’s description, but Luden's Tempest doesn’t seem to be part of the AP build.
• The MR threshold you name at Void Staff is 150. Did you do any calculations on that or did you generate that number out of thin air? It seems very high to me (for example, for Lissandra it becomes gold efficient to build Void Staff once the enemy team has at least 50 MR... ).
• Phantom Dancer is listed both as an offensive and as a defensive item (at both the AD and Muramana build).
• The Muramana build is a bit confusing, is building Nashor's Tooth classified as an AP build? Do you commit to AD when you buy Blade of the Ruined King & Death's Dance (& does this always mean building crit?)

The presence of this section is VERY confusing. It took me way too long to find the link to your matchup spreadsheet. You should definitely make it more clear that there is a matchup spreadsheet and make clear where the link is (aka not in colored BBcode so it practically becomes invisible). In the meanwhile I got distracted by the massively outdated information that is still present in this section. Perhaps it is a better idea to move the outdated stuff into a spoiler/separate draft till it’s updated?
This is simply a repeat of earlier sections. Notes:

• Not every listed rune is explained (what has Demolish done to you to deserve this)
• In the items you mention Luden's Tempest at Blasting Wand (first base), but Luden's Tempest is not listed anywhere.

Alrighty, I do not have much to say about your presentation. Your code is clean and your guide has a coherent theme, with one exception: the Summoner Spells. It looks really dated compared to the rest of your guide, perhaps you could modernize it/bring it in line with the main theme?