Only if Akali is really bad will that work. First, E-AA doesn't work once she has shroud up. Sure, the E hits, but you can't auto to proc it when she's invis. And since her shroud lasts twice as long as Illumination, you basically do nothing. Akali heals back more damage than E does with her first auto if she has even just a revolver.
Second, if Akali dashes to you with more than one charge, you're fucked. You snare her mid-air, and she stops right behind you. Now you either turn and try to burst her, which you can't because she is in range to heal off of you, or you try to run and she just dashes again when it's back up. The only way you can conceivably win is if you flash her dash and then combo. If you have enough AP, it should kill her, but if not, you're still fucked.
You can bully her easy peasy before 6, but the second she hits it, you're her bitch for the rest of laning. Same idea as Fizz, Zed, Yasuo, or any other melee into Lux. Once she has Gunblade, it doesn't matter if you have 4 kills on her, she will annihilate you. The only way you beat Akali as Lux is with ganks. Other than that you just look to win teamfights, since Akali is a garbage teamfighter.