Repair (rather than replace) 5th string peg?

An attempt to tighten a banjo 5th string, beyond where it had been before, produced a catastrophic failure of an old tuner. The square steel shaft simply came loose. With the end of the shaft no longer held in the plug base (? -- i.e., the brass thing glued inside the neck), tightening the screw on the knob produces no friction between the various parts. In fact, the knob and shaft assembly just slipped out.

The question is whether there is any way to make it work again, or is removal and installation of a new peg required?

I do not believe I lost any parts. The end of the square shaft has a tapered round part, with the taper angled to be held by something in the base plug -- now, presumably, worn and/or broken. That taper and the round little hole at the bottom of the base plug are visible in the photos.

If forced to remove it, I'm inclined to tap a thread into the brass plug, insert a machine screw until firm, and heat and wiggle. (There's a lot of visible glue; it's certainly not original equipment.)

(Please don't say I needed a geared peg anyway. It's a 19th Century instrument, and friction pegs do fine with light gauge nylgut strings.)