My Sansa Fuze won't turn on, even when connected to the computer

Your battery’s not charging. If you have tried the solution directly above your post, and it’s still not connecting, then either you have a bad cord or or a bad unit. 

If it’s actually new, take it back and get another one (complete with a new cord–or try it with a cord at the store).  If it’s used, you could try replacing the cord. It might be as simple as that.