What happens when my opponent plays Hidden Blade targeting my Akali, Deadly Weapon in reaction to her move trigger? Will it still deal damage to the chosen unit?

It will.

Akali is another case where her ability refers to information implicit to its own trigger condition, similar to Iascylla. Akali’s “battlefield I moved to or from” doesn’t require her to be on the board in order for us to be able to identify it, nor does it reference information about her current state. Even if she leaves the board, that information was checked when her ability triggered, and it is still available to us on resolution.