If you have +V on the C terminal then you should see +V on one of the others (NC/NO), flipping the switch should move +V to the other (NC/NO) terminal.
If it doesn't then the switch is dead (welded contacts?).
It should be easy enough to find another switch that will fit, but one of similar dimensions will likely fail in the same way (buy spares).
By the way, I avoid soldering on switches, the internals have an annoying habit of melting
Get some crimp-on blade connectors, it's also easier to replace the switch when it fails.