iPod Touch 5th Generation Home Button
The replacement battery I bought had contact strips that were too long and too wide for the slots on the circuit board, so I had to trim off about half the length and 1.5 mm of the width on each strip, but after that alteration, it worked fine.
I agree; in most cases the fault won't be the outer button but the dome switch underneath. For that repair the best source I found was here:
There are two very small dome-shaped parts, one of which is rimmed with orange adhesive Kapton tape that make up the dome switch; the lower one goes on first, convex side up. Then the Kapton taped upper dome goes on over that, with the small tabs in the tape aligned left and right when placed over the lower dome. This is a very delicate operation, and requires patience and tools for finely detailed work. I successfully got mine working again, so I can confirm that this repair is possible. The parts can be purchased by using the link provided in the YouTube video. Best of luck!