The light itself on the charger is not part of the charging circuit - it's just an indicator light that can be switched on and off and between green and orange by signals sent back from the MBP on the centre pin. It's frequently possible that the signal gets missed by the IC in the plug and it doesn't turn the light on, but charging continues normally.
Check what the computer actually thinks is happening when the charger is connected - does Activity monitor and the menu bar indicate that the system is charging, even if the light is not on?
The charger brick as a whole outputs a very low voltage when it detects it's not connected to the computer (if you put a multimeter on the pins you'll only see a few volts at most), but the handshake to the MBP switches it into high power mode and puts the voltage to 19 V (at up to 4 or more amps if the load is there from the machine). At this point it sends a signal to the chip in the adapter plug to change the colour of the light.
Possible failure points in this chain: the brick itself, the chip in the magsafe plug, the DC in board on the MBP, the MBP's battery (unlikely, but not impossible).
Check that the pins on the magsafe plug are straight and clean and not pushed in too far (the centre pin should be slightly longer than the outer pins), and that the pads on the magsafe are also clean and undamaged. The centre pin is what controls the light on the plug while the outer pins are the positive and negative terminals of the DC power supply (mirrored, so you can flip the plug over either way).
