Kassadin wasn't ever supposed to be an assassin.
TL;DR Kassadin was initially designed as a melee DPS champion but was played like an assassin because of game changes. I believe it'd be healthier for the game and beneficial to Kassadin for him to be retuned back into a more DPS oriented role instead of a burst assassin that can go out and in as he chooses.
As one of the original 40, Kassadin's kit and play patterns have changed fairly dramatically throughout the course of the game. Originally designed as an anti-mage Melee-Carry (got attack speed from taking magic damage, W gave mana steal and active armor pent, kit was intended to provide disabling and sticking power when hopping onto mages), Kassadin later shifted into a burst assassin after the removal of his mana burn, and upon the discovery that his AP ratios were really, really high.
Kassadin's original kit design was extremely oppressive when this playstyle shift happened, and led to him being basically a permanent ban at all times until his (unsucessful/misintentioned) rework. A point and click nuke with massive damage and a long duration silence coupled with a massive burst combo and unmatched in and out mobility wasn't good for the game.
His rework should have tried to steer him into his roots again, but instead essentially removed the silence and moved numbers around; this didn't work. The way Kassadin is currently designed, if he's good, he becomes an untouchable, instakilling monster. If he's not so good (as he is currently) he's basically trash.
I adore Kassadin's character, his theme, his appearance, and the original intent of his design, to be that guy that casters would always be wary of, watching for him to jump out of nowhere and murder their faces. And I would love to see him be useful without being so restricted in what he's allowed to have.
So I propose that his kit become a melee DPS one, with a set of abilities focused on utility and spell disruption that allow Kassadin to punish those who'd wield magic against him. Below is an example kit of what exactly I imagine:
Passive: Kassadin takes reduced physical and magical damage from champion spells and abilities. This percentage starts off lower, and increases at levels 6, 11, and 16. (An example would be 5/10/15/20%.) Kassadin's passive should serve to increase his durability significantly (though less so early, Kassadin's laning phase should be relatively weak for what he becomes) against spell damage that would otherwise blow him to pieces if he wanted to enter a fight in close quarters. Most melee-DPS type champions have built in defensive steroids (i.e. Yi's Q/W, Fiora's W/R, Trynd's W/R, Yasuo's Shields and Wind Wall), and Kassadin's should be more specialized.
**Q: **Straightline Skillshot (gasp!) that does a rather low upfront amount of magic damage (has minor AP and total AD scaling), and charges the victim with negative energy for a few seconds. If they cast spells during that duration, they take additional damage and are briefly silenced (imagine .5 seconds or so, the major intent is to be disruptive to spell combos.) -Kassadin's Q was intended to make it harder for casters to blow your face in if you hopped in on them, but the original silence was somewhat oppressively designed (point and click, high damage nuke, long ass duration, attached to a champion that can gapclose at will.) As a skillshot, it can be missed or juked, can't harass through minion waves, and isn't as easy to farm with. The "feedback" effect aims to make it more consistent and interactive as an anti-caster tool that's less choking to play against. You can still cast, but you take extra damage for doing so, and it's difficult to chain spells (note: it not being a straight up silence also means you can flash/Zhonya's/use an escape even if it's landed, his original silence didn't allow this and was one of the reasons it was so strong.)
W: Passively increases attack speed per point, and adds a percentage of Kassadin's total AD and AP as on-hit magical damage. Actively, doubles the attack speed bonus for a moderate duration, resets his attack timer and causes his next basic attack to do double the on-hit damage. -This is where Kassadin's damage is coming from. A powerful double steroid, this gives him a window to dish out some serious hurt if allowed to do so, with a spike of damage to let him (at melee range!) chunk a target down to size. Attaching the attack speed boost to a temporary steroid allows one to bait it out or CC him when he uses it.
E: No longer runs off charges, instead, nearby spells shorten the cooldown (including Kassadin's.) Does lowish damage in a cone (again, total AD+AP scaling) and slows for a short duration. -I envision Kassadin's E as a tool meant to assist in sticking to targets, the damage being secondary, but instead it ended up becoming one of his primary burst nukes. The charge system it currently uses, while unique, is inconsistent and varies greatly in usefulness depending on the compositions of both teams, meaning in some games Kassadin rarely gets to use it unless he wastes effort charging it, and in others, he almost spams it. Making it cooldown based adds more control over the ability and makes it more reliable.
R: Passive: Kassadin returns mana equal to a percentage (increases with rank) of the damage dealt by his basic attacks and W. Active: Kassadin blinks to a nearby location (I imagine the distance being a middle ground of his pre-5.4 range and the current range), fading partially into the void for a brief (.5-1 seconds), removing and granting immunity to slows during that duration. If Kassadin teleports nearby an enemy champion, the cooldown of the spell is massively reduced. Each time this cooldown reduction occurs, the mana cost increases by a flat amount up to a cap. The duration of this increase is equal to the spell's base (doesn't include CDR) cooldown. Does not do damage. -Aight so, Riftwalk. It's problematic. It lets Kassadin be pretty much everywhere. It's primarily meant to be a sticking tool, but the way it's currently implemented it's a pretty-much-everything tool. With this implementation, I aim it at being really good at the sticky part, and not so good at the other stuff. The cooldown when used outside of combat (or when used to run away) is rather hefty, making it less obnoxious to play against. When used offensively it's a lot shorter, but it's still mana-gated after a point, Kassadin's only real mechanism of sustaining it being auto-attacking.