What is busted about the jack? Can you ohm it out still?
Can't think why a whammy would use switched jacks... I don't think any of them used batteries as the had AC supplies for v1-4 and ran 3.3v for the digital and 12v internally via one half wave and one full wave rectifier, again iirc.
Check the traces/ohm the pads and see if the pins connect via trace. Ohm each side of the jack and if ring and sleeve connect.
I'm betting that it's a modified stereo jack and only utilizes tip and sleeve, hence ring being cut on one side.
Or ring and sleeve short/connect on the pcb.
If that's the case, just use a stereo jack.
The only other thing I could think of is, if one side is switched, it's switching ground since they are AC input only(v1-4) so ground is established via the jacks, unless it's running on a crazy virtual ground network.