I had my slim acting up on me a few months back. I would be playing a game and it would just freeze up and turn itself off completely.
Have you tried to get to the reset menu? I know you're saying that the power button doesn't work all the time, but here's how. Press and hold the power button, you'll hear a beep that turns the unit on followed by another beep (which I think is the video reset) and then two quick beep beep. Let go of the power button. It'll tell you to plug your controller into the PS3 with the USB cable.
You can test with any of the options you see there. The factory reset will obviously wipe messages off your system and will make everything default again.
With my slim, the issue ended up being the hard drive. I restored the file system, rebuilt the database, and did a full system retore. Nothing worked until I ended up buying a new drive and swapping it out.
You didn't say if the system was on when it fell. My theory is that you might have a bad connection with the power button as well as a faulty hard drive. If you have a spare laptop SATA drive, I would put it into the PS3 and see if it works after formatting the drive.