If the battery itself is working, it may be either the battery door or the SD card door. Each has a small switch that if not activated when the door is closed, the camera will not operate, and appear "dead". My camera had a small plastic "pin" on the door designed to activate the door when closed and this pin had been broken off.
You can check the switches if you know the battery is good. Just turn on the camera with the battery inserted and use a small screwdriver to activate both switches. Keep in mind that both doors have to be shut with the switches activated to allow the camera to turn on.
This is a well known "issue".