I've had the same exact problem. It seems that your DC jack input got disconnected from the motherboard ,preventing power from transfering to the Battery.
I just bought an External Battery charger to solve the problem.
While an external charger may solve the problem so long as you want to work from a battery, The problem you have is the A.C. connector on the mother is shorted. That is why the power supply shuts down. it Will not shut down on an open. There may be something stuck in the jack, or a diode or some other component on the motherboard that has shorted. The best way to check this is with a Volt/Ohm meter and trace for the short.