There is a switch in the battery compartment that shuts the camera off, when you open the battery door. I know it's a long shot, but see if you can find that switch and activate and/or deactivate it with something. It might be jammed.
My wife dropped her PowerShot and had a similar issue. It would no longer power on, so she bought a new one. A few months later, when I had some time to play around, I got a tooth pick and used it to probe the battery door switch. It had somehow gotten jammed. The toothpick freed the switch and the camera sprung to life. It is still working today.
Like I say, it's a long shot, but it's worth a try. I could certainly see something similar happening based on the description of your issue.