Well, even broke, as long as it will turn on, you should be able to pull data off of it. You would need to plug it into a computer to access the device, like you would any folder on the computer itself.
From there you should be able to copy and/or move the files, etc., over to the computer, so that you can then put them on whatever new device you get.
This, of course, is important if they replace it or wipe it when repairing. If they just repair the device, and don't wipe it, then your stuff would still be there.