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?