If shorting the MOSFET made it work normally the fault has to be in the area of the IC. Any of the parts around there could be faulty, it's difficult to make a diagnosis remotely.
If the MOSFET is working, there is a test you could try but it isn't conclusive: measure the voltage between source and gate pins of the MOSFET, it should be >7V when the power is on. if it isn't, either the MOSFET has an internal gate to source short (unlikely) or the IC could be at fault. 4013 flip flops are very inexpensive and easy to find.
Brian.