Illaoi is far from broken. She's only "broken" against players who don't know what she does, or don't know how to move. Trust me, she's not even close to being broken or OP. If anything they nerfed her really hard so she's in a meh state right now.
Key tips I can give you to help lane against Illaoi:
1: Move.
If you don't move then you're giving her easy setups for her Q and ult. Literally move out of her Q. It's a very slow moving, windup skill shot that can easily be dodged moving left or right. You wont get hit if you move. Same thing with her E. just stand behind minions and she can't get you. If you do get hit, then fight her off because that can easily prevent you from becoming a vessel.
2: Illaoi has terrible mana regen.
She can't spam her abilities because of her small mana pool, and relies a lot on her tentacles to do most of her damage. If you see her OOM or low mana, then that is when you go in. If you go in when she's full on mana, then you're going to have a rough time.
3: She has no CC or gap closers.
No, her W is not a gap closer. It gives her some distance, but trust me it is not reliable and still requires you to get very close to the enemy. Use this to your advantage if you're playing a ranged champion. Poke her and deny her farm. As long as you dodge her tentacles you'll force her out of lane since she has no way of getting close to you.
4: Positioning.
Never, ever hug the walls against an Illaoi. That's asking for a bad time. If you have to fight her in lane, do so in the middle of the lane. Never against the walls unless you need to (Gnar, Poppy, etc). The further away from her tentacles she is, the more vulnerable she becomes.
5: If she ults... MOVE.
Never stand in her ult. That is the dumbest thing anyone can do. She has no way of keeping you in there either. So just run out, wait it out, and then go nuts because by then she wont have much left to kill you with.
6: She's a lane bully, but terrible in team fights.
Yes, she is extremely strong in lane. But this is only because of walls. Without walls, she can't make tentacles. Without those, she loses a lot of damage. She also has no utility outside of E (if she lands it), so her team fighting capabilities are very limited. So use the laning phase to farm up CS. She doesn't scale well so by the end of the laning phase you'll be able to duel her in team fights if you play smart.
Reason she's also bad at teamfights is she requires specific positioning in order to get the most damage out of her ult. Why? Because she needs walls to make tentacles. If you fight her in mid lane she can't do much, but fight her in close quarters (like the jungle) and she will wreck your shit.
Trust me when I say she's really not that hard to go against. Once you learn what she does she's really easy to beat. Another way to beat her is to build
or
. Or even taking ignite can really hurt her health regen form her tentacles. But it's like I said, as long as you move and play smart, she is easy to beat.