Well between your messed up digitizer problem and now the sync button, it seems you've got a few bugs - although apparently paper around the edges fixed your digi.
Obviously, then, you either send it in, replace it or do without the button. If the icon works you could keep it, and there are alternative programs which allow you to activate hotsync through a penswipe or other means of "launch," so it's really up to you. Not to overstate the obvious, but it's really your decision if you have had both a digitizer and now a hardware (button) problem.
Try not to drop it again! Although, that might just fix the thing . . . . a buddy of mine had a camera in for repair and subsequently abandoned by the customer - which refused to work until he lost his temper and slammed it into the countertop 3 times. It's worked ever since and photographed a trip to Alaska!