I've been having a hard time finding other people with this problem, but the few I did said that sending it to MS (and getting a new one) is what they had to do. One person did say that they were able to fix the problem with a bootable Surface 2 recovery USB. I have a lot of work on my surface that I don't have copies of, and would really like to be able to save those files.
Things I have tried: Holding the power button with a combination of volume rocker/home key. Holding the power button for 20 seconds followed by 2 seconds off and on. I have fully charged it and know it has enough battery.
Help would be greatly appreciated.