How do I transfer my game progress to a new Apple ID?

I tried searching but only found ways to transfer data to another device, not another ID. I started playing Cooking Fever with an old account (now belongs to someone else), whenever updates are required, my mini iPad prompts a password for the old account. The old Apple ID is erased from the device I'm currently using, only ID in this device is the one I'm using now. I don't want to keep asking the current owner of the old Apple ID for the password as said owner frequently changes password.

All help is appreciated.