If you do a hard restart (hold the home button for 5 seconds, while continuing with the press, push the volume down for 5 seconds, keep holding and then top button until it restarts) that should work. If not plug it into a computer, then press the top and home button for about 10 seconds, then release the power button while holding the home button until the computer detects an iPhone in restore mode. Restore the device and it should work.
I have fixed these clips before on an iPod nano... They are EXTREMELY difficult to replace the parts from another clip. I know that there are some people who can desolder them and solder on new ones, but it is incredibly difficult.
This looks like something that a screw goes in to. I wouldn't worry to much if you are missing a screw, but this is what holds in the screw, and might prevent it from being shorted out. Make sure that all your screws are tight, and if one isn't that is where this goes.
You could solder it to the logic board where the 5v pin is on a USB port... However this would take power away from the USB port. It may be possible to take it from the keyboard itself since it is a USB device. I would check the ribbon cable with a multimeter for the right voltage. Another place to check would be the disc drive power connector. Depending on how often you use the drive, this might be a good option. I'd like to here more about this mod, since I hate typing in the dark. Maybe put up some links to some images.
I am having this problem as well. I received it torn apart with a broken hard drive. I replaced the hard drive and did a restore, and the click wheel and buttons don't work. I replaced the logic board thinking that was the problem, but that didn't fix anything. The hold switch isn't the problem because the icon will go away. I believe that this is actually a scroll wheel problem, and would be a simple fix. I am waiting on a new one, and will respond with the results. Edit: - The new scroll wheel fixed the problem. I tried a 5th gen scroll wheel, and that didn't work. There must be something unique about the classic's wheel.
I don't know if it is the same as the iPhone 3G, but my iPhone would do this all the time... Try holding down the home button for 3 seconds, keep holding it down and push the volume up button for 3 seconds, keep holding those down and finally press the top button until it turns off. Press the top button, and it should come back.