I know I am a bit late on this but I had a very similar problem with my Cd/dvd rom that came with my mini. It would just spin and spin and windows would not recognize it. I called dell and told them it was dead and (after much fussing with the woman at tech support) they sent me out a replacement drive and it has worked marvelously since then.
She tried to run me through hoops though, kept telling me that the drive wasn't compatible with my mini (which is BS), tried to tell me to send it back and then go to my nearest dell store and buy a drive there (don't have a nearby dell store). I eventually had to get kind of irate and just insist that she send me a replacement drive.