My k3g hanged about a month ago when I was reading using Duokan. The screen stayed still and no matter how hard I try to restart it or reset it, there were no response except for the LED. This some how seems a little similar with your case.
My final solution was to take the battery off which shuts it down completely and then starting it up.
Edit:
OK, I guess your's is different as it is able to proceed to the "Your Kindle Needs Repair" screen
I've also encountered that once and a factory reset solved it.
Good luck.