Honestly since the time stuff is a bit of a mess the easier explanation is because Aku jizzed some of his essence into a cup for crazy ladies to worship and one of the crazy ladies used it to make a half-Aku baby, then with Aku dead Ashi's Aku essence has been slowly burning away inside her.
In a kinder series when this runs out she simply loses her mystical abilities, so all the Aku powers she doesn't need not anyway. However it could be argued, if this is even true, that the series chose the ending where half a being cannot exist on its own. With her Aku half destroyed she simply can no longer continue existing. However all of her actions in the timeline still happened for the new future to be created and this explains why Jack still remembers her. She still existed, she just cannot exist anymore.
Now maybe, MAYBE, he can find a way to bring her back using magic (though I suspect Genndy prefers the ending where she's gone but Jack carries on and "you gotta carry that weight"). If we absolutely need Jack to have love in his life then, okay, the Jack we saw thrown into the future arrives in a scifi paradise where Aku was defeated, stays for awhile, meets the Scotsman and everyone else and they're all super happy, no problems, and then he eventually returns to the past ... only to discover the other Jack who arrived with Ashi. They stare at each other for a moment, then smile and shout, "NOW NEITHER OF US SHALL BE VIRGINS!"
Boom, there you go. Happy ending.