Okay, now bear with me here...
I understood the assignment, and heat shrink is just beyond my capabilities right now, but here's what happened:
I managed to pull the wires out of the old plug and found this.
When pushed in all the way, this is the only metal that connects to the fork on the tinybox.
It's tiny, too, so I doubt seriously my ability pry it apart and reuse it, but its clear where the solder goes, at least, and I know I can do that part.
What is this meta piece called and where would I look for it?