Inverters, like most electronics, typically have two states: working perfectly fine, and suddenly not working anymore.
Some internal component fails, for whatever reason, and then nothing happens when you plug it in.
So the bad news is that if your car power inverter suddenly stopped working, there’s a pretty good chance that it’s just broken, and it will probably be more cost effective to just buy a new one.
The good news is that there are a few things you can check before you throw in the towel.