I'm actually relieved to see this thread. I'm in exactly this situation, and I never expected it. I know that the schools will take the letters late, but this professor is writing the most important one. She's also been incredibly supportive both on the the phone and by e-mail about my application and agreed in July to write them. December 15th was the deadline for a program I'm really interested in attending. That deadline has now sailed right on past. I don't want to send any more e-mails, because I was starting to feel like a really chipper stalker. So now I'm scrambling for a replacement for my remaining apps. and hoping that she at least sends the LOR, even late, to the other program. Yuck.
Good luck to everyone waiting on letters!