Your battery clearly needs a charge or replacement!
I would review this Apple T/N Apple Portables: Troubleshooting MagSafe adapters
Make sure you are using a REAL Apple charger! There are a lot of fakes! These fakes can damage your system in a few different ways. The first is damaging the MacBook Pro Unibody 13" and 15" MagSafe DC-In Board carbonizing the gold contacts so the connection with the MagSafe is not being made. From what you describe I’m suspecting that’s your problem. Here’s the guide you’ll need to follow to replace it MacBook Pro 13" Unibody Mid 2010 DC-In Board Replacement and don’t forget to get a new charger! Here’s what your fake charger did Lacking safety features, cheap MacBook chargers create big sparks
After replacing the DC-In Board: If your MagSafe charger is Green then your logic board has a problem. If it’s Amber then you need to let it charge the battery for a good 10 hrs! Then see if it starts. IF it does download this gem of an app CoconutBattery. Take a snapshot of the main window and post it here for us to see Adding images to an existing question.