Not a fix, but a workaround:
1) Remove the outer button and use the plug of the dongle to push the inner button to turn the ipod on or off
2) When the inner button falls off or stops working, remove it. You will be left with just the base contact for the button, which looks like a small round metal contact separated into 4 quarter sections. Now you can lightly touch the plug of the dongle to the contact and move it around a bit to close the contact between any 2 of the 4 sections in order to power the ipod on or off. It only takes a couple of seconds to do the trick and the inconvenience may be worth it compared to forking out another $150 for a new ipod.