I would find out what brand the faucet is and go to a plumbing supply house and get a replacement handle. This not a rare problem, so you should be able to get the part. Replaceing the faucet entirely could cost a few hundred dollars. No need to tear up your wall and solder on a new expensive fixture if all you need is a handle.

I would pull off the handle and try to figure out what brand it is before you bring it into the supply house. hopefully you will get a patient person who will find it in his books.

Well, now that's the problem. It's a Delta. But I have pushed and pulled, twisted and tapped (all the while of course looking for hidden screws or removable covers) but I'll be darned if I can get the handle off. I can remove the little water temperature adjuster on the front via the screw down the center; but I can't get off any more than that.

http://www.deltafaucet.com/wps/portal/deltacom/kcxml/04_Sj9SPykssy0xPLMnMz0vM0Y_QjzKLd4439vIBSZnFW8aHOepHogtZIoSC9L31fT3yc1P1A_QLckMjyh0dFQEAPpFv/delta/base64xml/L3dJdyEvd0ZNQUFzQUMvNElVRS82X0NfM0pT

Use the 800 number or Chat.