Is it okay to leave circuit on when a bad GFI has tripped?
You can leave the circuit on. The bad GFI just won't provide downstream power is all. There's no hazard there.
The GFI is bad and needs replacing. If you're OK turning the whole circuit off, you can use a screwdriver to replace this easily. The main thing you want to do is take note of which wires are on the LINE side (as in do this before you disconnect the old GFI). Mark them with tape and then disconnect the LINE side, and then the LOAD. When you hook the new GFI up, connect the LOAD and LINE just like they were in the old one. It's important you not mix them up. Beyond that, it's literally just unscrewing and screwing.