First, check around the Apple.com forums, it’s possible there’s a fix for this that I just don’t know about. Secondly, if you have an Apple store near you then you should just take it there. If this is a simple hardware fix you’d be better off letting them just do it.
Failing all that, plug a firewire cable into each Mac. Turn on the mini and hold the “T” key on the keyboard.
That SHOULD load the mini onto the other computer as an external firewire drive. You can just copy whatever you want off of it. (Now, since I don’t know for sure what’s wrong, maybe this won’t work. That’s how it’s supposed to happen, though.)
After you do that I would simply try erasing the hard drive with the OS X CD and then re-installing everything. There’s a good chance that will clear everything up.