I opted to bond the two pieces of pad back together with a large plastic sheet (icecream lid). I scored the top of the pad and plastic sheet to allow for good glue stickyness (I'm not sure it acually helped any.

I glued along the seam first

Then I attached the plastic sheet to the top. The glue didn't spread as well as I would have hoped.

Then some additional reinforcement on the back. I smushed the glue better on the last one.

Then soldered a new connector to the piezo. I think the original was a long cable, but it didn't seem to pull out enough for me to solder to it directly.

I don't know what it is supposed to look like, but the voltage spikes seem to match the tapping. It seems to work (at least some of the time) in game. So, success?