Mine is worse. When I plug in the charger nothing happens. No battery icon. No nothing. Leave it charge for 10 hours and still nothing. Same charger meanwhile charges my second Nexus just fine. When I try to start it, nothing. If I either hold the on button or a combination of on button and volume down, I do eventually see the battery icon come on and off at intervals. But there is not lightning bolt in it and I assume it is not charging. The unit will not boot the OS regardless.
I have a spare battery for one, so I may just pry it apart, put the new battery in... install Helium.... back it up... and restore everything to another Nexus 7. Does anyone have any ideas I should try first?