I have the same problem with the J1. Battery is good, conatcts are good still won’t turn on. Most likely the power board (the PCB on the top) has an issue. I have already measured the Zener diode at the power input and the fuse. Unfortunately I could not find any schematics for further troubleshooting so I think I will replace the whole power board if I can find one at a good price. If I wanted to have it fixed by Nikon I would need to send it abroad as they don’t have any repair center in my country.