I'm having the exact same problem.. and I'm pretty sure that mine had something to do with an update around that same time too. Surface 2 has been in the same spot on my same desk for 5 months without being moved, so I know it's nothing that I could have physically done. The Windows logo still works as well and I did everything listed on Microsoft's troubleshooting page INCLUDING a full-recovery and it still isn't working. Figures b/c my warranty ended at the end of May.
However, one of the times that I turned it on after my battery died, it did work. Not for long (in fact I think it only lasted until the Surface fell asleep again), but it did work fine. I've read other forums where many people are having this problem and some people are having success draining their batteries and things of that nature, but it doesn't always work. So after 20 hours of looking for a damn solutions, I haven't found a real long-term solution.
Another common claim on the forums is that Microsoft will not do anything to assist. If they're not going to help me with one of their screw-ups, I'll wreak hell upon them. But for real, if they do not help me tomorrow, they'll be giving all Surface 2 owners a free replacement or upgrade when I'm done with them.. I'll let ya'll know if I get lucky and actually get an answer out of them.