That's not much circuitry but I would say it has to be a timing circuit. If the LED is flashing it should be working.
Your white wire goes to a device that I am guessing is a transistor. It is also the only thing heavy enough to drive the transformer.
The best way to test a transistor is a transistor tester. That one should have a crossable number on and MIGHT be one you could grab at a Radio Shack. (But you never know about them.)
If that is the problem it might need more heat sinking.
RT (definitely my 2 cents worth)