There is a part inside the computer behind the power port where magsafe plugs in, commonly referred to as a 'DC-in board' and the circuit in there may fail, or partially fail, so the connection may work part way or not at all. But it may not have failed...
Is the surface of the metal areas where magsafe touches, appear clean?
There may be a way to reset the System Management Controller to see if that may help the computer see the new battery and charger, since there are times when the issue may be elsewhere in software, or firmware.
•Intel-based Macs: resetting the System Management Controller (SMC)
There also are Support articles on Calibrating a battery, among other things.
The DC-in board is a repair item, see http://www.ifixit.com/Device/Mac_Laptop section on macbooks to find your model to see what is involved, but like most things, to properly identify which MacBook you have since processes, part #s, & steps to fix things vary. It may involve a repair tech + new part, if you aren't into that.
Good luck & happy computing!