Reset the AirPods first (not sure if you already did that). Hold the back button while the case is plugged in until the LED flashes amber. Keep holding the button until it flashes white.
If the LED doesn't respond, then the button itself is probably broken. And that's not going to be easy (or probably even possible) to fix. So, I would recommend getting a new case.
I am not a 100% certain that the case for AirPods Pro 2nd Generation or AirPods Pro 2nd Generation (USB-C) works with AirPods Pro 1st Generation. You should probably check into that before purchasing the case.