You've tried all the appropriate stuff, but you might try the firmware again and/or leaving the battery out a full 24 hours. Also see if it works on the dock, if it works fine, I'd conclude that the battery is the problem. If it doesn't, that still doesn't necessarily eliminate the battery. Obviously it's not the screen itself, if it's not the battery, possibly the dock, or the firmware, it's not something you can fix.

Since you indicate that it worked OK for a short while after removing and reinstalling the battery, my guess is you need a new battery but it's a slight possibility it's the dock.