Akali plays right into Lux's Q. If you can spot when Akali is gonna try and jump you, you'll be fine.
Lux shouldn't have too much of a problem with the likes of Akali, especially if you get farmed / fed. The biggest assassin's she has a problem with are the teleport-happy ones (Talon, Katarina, Zed). It's important to keep your head on straight when someone like Akali, Diana, or Kha'zix comes at you. Most of the time, it's engage-or-escape for them. They can rarely do both, so if you don't panic and play smart you can turn tables on them.
Specifically against an Akali, you want to watch out when she hits 6 (since that's when she'll start jumping at your face) and when she completes her Hextech Gunblade. Both of these events are power spikes for her.
And always remember that the tower is your friend and people underestimate it's damage and range when they get that tunnel vision on a low-health champ. W as soon as they move in for that jump. As soon as they come at you, get that binding off. That'll be two or three tower shots they eat, and your W will be coming back around the second shot. Can't tell you how many times I had this exact situation play out.
Mostly it comes down to harassment, not panicking when they ult onto you, and learning to spot the enemy's tell when they're gonna jump.