What abilities do %health damage?

Solution 1:

Amumu's Despair deals % max hp in a small AoE around him over time as long as the ability is toggled on
Brand's Blaze passive deals % max hp when he deals damage with any of his spells
Dr. Mundo's Infected Cleaver deals % current hp
Elise's Neurotoxin deals % current hp
Elise's Venomous Bite deals % missing hp
Evelynn's Agony's Embrace deals % current hp in a targetted AoE
Fizz's Seastone Trident's passive deals bonus damage % missing hp on basic attacks
Garen's Demcian Justice deals bonus damage based on target's missing hp
Jayce's Thundering Blow deals % max hp
Kog'Maw's Bio-Arcane Barrage deals % max hp on auto-attacks as long as the ability is active. Also increases his maximum auto-attack range
Malzahar's Null Zone deals % max hp over time in a targetted AoE
Mordekaiser's Children of the Grave deals % max hp over a long duration. If you die while affected by this, you become Morde's "pet" for a limited time.
Nasus' Fury of the Sands deals max hp in an AoE around him as long as the ability is active
Poppy's Devastating Blow deals bonus damage based on target's % max hp
Varus's Blighted Quiver (upon activating the passive marks) deal % max hp
Veigar's Primordial Burst deals increased damage based on target's missing hp
Warwick's Hungering Strike deals % max hp

Jarvan IV's Martial Cadence deals % current hp as bonus damage on his basic attacks
Kha 'Zix's Taste Their Fear (With Evoled Enlarged Claws) deals bonus damage based on target's % missing hp to Isolated targets
Lee Sin's Resonating Strike deals % missing hp
Nidalee's Takedown deals increased damage based on target's missing hp
Riven's Windslash deals increased damage based on target's missing hp
Vi's Denting Blows passive deals bonus damage based on target's max hp
Volibear's Frenzy active deals increased damage based on target's missing hp
Xin Zhao's Crescent Sweep deals % current hp in an AoE

Vayne's Silver Bolts deal % max hp every third basic attack against a single target

Blade of the Ruined King's (Not available on the Crystal Scar) passive deals % current hp on basic attacks
Blade of the Ruined King's active deal % max hp and also heals the user for that same amount. Effect also lowers your move-speed and boosts the user's

Blackfire Torch's (Twisted Treeline and Crystal Scar only) passive deal % max hp when you deal magic damage. Uses 1 charge per second and contains 18 at once, which are restored over time when not used.
Deathfire Grasp's active deals % max hp and increases magic damage done to that target for a limited time
Kitae's Bloodrazor's (Crystal Scar and Proving Grounds only) passive deals % max hp on basic attacks
Liandry's Torment's (Summoner's Rift and Proving Grounds only) passive deals % current hp when you deal magic damage

In addition to all of these, Mikael's Crucible's active heals a single ally for a % of their missing hp