I noticed you don't weave any AutoAttacks in your combo, which might explain your lack of damage.
Akali's Q applies a mark to the target, which can only be detonated through AutoAttacks.
E used to do it, as did R, but they removed it from E and R places you out of range for it to do the free AA it has coded.
So as it stands, this is her current combo:
Q
Wait for Q to land. (R is faster than Q)
AA
Q again (low cooldown)
R
AA
W if necessary to wait for cooldowns further, and use Hextech if you need to guarantee AAs (it slows them).
Do not use E when trying to burst someone down. It doesn't do much damage and it costs too much energy. It's best used for waveclear.
The biggest part with playing assassins though is knowing when to engage on someone. Knowing this is 10x more important than knowing your combo.
There are two things to remember as an assassin:
1- You suck at teamfighting. You're a glass cannon, so if you get caught even for a second, you die. You're better killing targets in smaller fights.
2- If you're forced to teamfight, you cannot initate practically ever. People save their crowd control for you, and you have to wait until someone else soaks that all up before you go in.
This is why I personally don't recommend assassins to newer players - you need to have knowledge of most champions to play them effectively. You're already invested in playing her though, so don't abandon her. Once you understand how to use assassins well, you will be a monster on her.