How to diagnose a broken transformer?
So what happened?


Well, the rule with all things electrical is, they work right up until they don't. With a small transformer like that, it is likely if it overloaded it could burn a wire in half thus failing open.

Apparently, you have a basic voltage tester so you can test for continuity (or resistance) on the primary side (240v) of the transformer, and test the secondary side (120v). It is possible some part of the external wiring  has opened and could be repaired. If either the primary or secondary are open, internal to the transformer, then you could replace the transformer. If this is cost prohibitive then, unfortunately, you may have to buy a new one. 

Good luck!