Battery Not Charging and Light Not Showing Up Again
The first thing you should do is examine and clean the MagSafe connector, both on the cord and on the machine. MagSafe's have five spring loaded pins; if one of the pins is stuck, or grime is blocking a connection, that can cut the power output of the cord, meaning the computer will run but the battery won't charge. Apple says:


  To clean the MagSafe connector and pin area, disconnect the adapter
  from the wall outlet and from the computer. Remove debris gently with
  a cotton swab or a soft bristle toothbrush. Be careful not to get any
  cotton fibers stuck in the pin receptacle and to not bend or otherwise
  damage any of the pins. (see: MagSafe Troubleshooting)


Failing that, this likely needs repair.

The MagSafe board is inexpensive and relatively quick to replace. Next is the battery and all cables, most expensive and difficult to exchange is the logic board.

You would want to replace parts one by one until the hardware issue is resolved. 

Resetting the SMC is a once every 6 months thing, if you do that more often it’s hardware repair that’s needed.