There aren't a lot of options for reasonably-priced, small capacity inverters. They're pretty much all chinese crap. I'd just buy a new one.
There's no blade fuse in the actual unit? Like on a car amp? I would expect that. Or a glass fuse inside the cover.
Did you test the output of the unit with a meter? It may not be zero; you might see low voltage, which would likely indicate something wrong in the step-up transformer portion.
I'm no inverter expert, but I'd suspect two common failure points- the mentioned transformer generating the high voltage, or the transistors used to convert to AC. I'm assuming you've probably got a simple square wave unit.
If you don't see something obvious like scorch marks (can be on the component side of the PCB, or underneath with the solder joints and copper traces), I'd test the windings of the transformer with an ohm meter, and then check the transistors (google it- different types are checked different ways).
Also, examine any and all capacitors for buldging tops. A visual check is often enough for those.
If you make it to the small bits, like checking individual diodes and resistors, I'd just pitch the damn thing and get a new one. Cheapest parts, but least failure prone and often tricky to replace.