That would be fairly difficult…. what I might do instead is remove the prong assembly and solder a short appliance cord to the board where the prong assembly wires go. Most of the short cable could then be tucked into the box and your device could keep working.
That’s a tough one. If you were able to remove the prong pieces from the assembly and had a friend that was gifted with a TIG welder…
That’s sort of the catch-22 of taking stuff apart and repairing things. You’ve got to do enough of it, and break some things, in order to figure out how to get these types of things apart without breaking them.
I don’t carry replacement batteries, or sell anything of the sort, but I’ve seen lots of LiPoly packs in various shapes and sizes for sale on ebay. I’ve had a lot of success ordering parts like that.