Akali only beats Diana when Akali chooses to get a defensive item and Diana does not, assuming similar skill and comfort on the champion.
When this happens, Diana has to spend more mana against Akali whereas Akali has sustain and no mana limitations. Akali can now dive Diana with 3 ult charges and win due to her damage being sorta hybrid while Diana's is all magic and is dealt with by the MR.
Other than that, there isn't any point in the game where Akali can reliably duel Diana unless Akali cleverly evades the Diana Q with her W. Then Akali can all in with ease. Even at 6 where Akali gets all charges for free, she won't have the damage yet to burst Diana through W before Diana QRRW Passive kills her. Which it very nearly will unless she bought MR because a 330% AP ratio (plus ~500 in base damages) delivered in the span of 1-2s is ridiculous. It -shouldn't- outright ohko but her passive's sustained damage will (or force the fight to end).