I checked lolwiki for the number. At lv18, It seems kat has higher damage number than akali.
kat full combo: e-->w-->q-->r, doing 1195 magic damage
akali full combo: r-->q-->e-->r, doing 950 damage. With her passive, she probably can do roughly 1000 magic damage.
I only play akali a few times and have never played kat.
My guess is kat probably would win if the fight is outside the akali's shroud. If the fight is inside the shroud, it would be an even match as Akali get 50 bonus mr from shroud.
However, it's just a guess based on a roughly calculated number.
PS: just an interesting thought, Akali get 50 bonu mr from shroud. So, the damage from kat's ult is (60-50)*10=100 magic damage, or 60*10-50=550 magic damage. ehhhh, This is really big difference.