If you have a multimeter, you could try measuring the AC voltage across the tweeter when playing music. If not, you could buy a dirt-cheap speaker driver from Amazon (you should be able to find something for under $5). First just try to wire it in parallel with the tweeter.
If you are getting a signal from the new driver, it's likely there is a broken connection inside the tweeter which can happen if too much heat was applied when soldering the wires to the tweeter.
If you aren't getting any signal from the new driver, disconnect one of the tweeter leads and try again. It's possible (although unlikely) that the tweeter is shorted, which would prevent a signal from the new driver as well. If you are still not getting any sound from the driver, it is probably a bad solder joint in the crossover.