I don't know about Ashi being Aku's legit 'daughter' but perhaps the Cultist leader meets up with Aku and Aku gives the cultist leader something to allow him to take control of Ashi.
Long story short, Ashi gets possessed, Jack is ultimately bested from a mix of damaged morale and Aku making use of the remaining Daughter of Aku (who didn't turn against him), the cultist leader, and Scaramouch (all three still end up dead at the end of the fight though) and Jack loses the sword to Aku.
Final battle happens, Jack, pissed off Aku fucked with so many of his friends, rips Aku a new one but Aku points out he'll just revive himself through Ashi, forcing Jack to do a 'dive to soul/mind'.
Jack tries to thrwart Aku within Ashi, but ultimately can't and remembers his own experience, instead, he manages to give Ashi the courage and support she needs to drive Aku out, leaving Aku and shriveled, black squid on the ground for Jack to vanquish once and for all!
At the end of things, many point out that while Aku is defeated, there's still many problems left and Jack has no way home.
Jack just smiles and says "I am home, and Aku was not just the demon we fought, but the misery he caused and I will continue to fight him here too"
The ending shot shows Jack in a new outfit that's a mix between his armored look and the robe looking out over at a village in clear need of help, soon joined by ashi in a similar look, they smile to eachother and jump onto some sick-ass motorcycles and ride down to save the day. (not confirming if they've hooked up or are remaining evil fighting buddies)
Also, I'd let jack keep the beard, though maybe trim it.