The problem is not the battery if you hear no startup sound. It is likely a bad connection with the ROM or something else.

I could be mistaken but the screwdriver in the picture looks like a phillips (cross shape), and the screws in the Mac are Torx, which is a 6-point star shape. "T-15" to be exact, but you might be able to get away with T-10 (not recommended, it could strip).