There are mainly 2 possible problems.

The first of them is a brake in the internal charge circuit. Do the following test: remove all batteries and attempt to power the laptop using only AC power. If this doesn't work it may be the problem stated above. Here is one way to solve it if you have lot's of guts to do it. Dissasemble the pc and look near the ac power jack. You should see a small chip (about 8 pins - 4 each side) , check to see if the pins on each side are linked together, or at least 3 of them are linked in this manner. If so, look further and you will see a group of resistors linked at one side of the chip and to some other components. Attempt to place a resistor (say 100kohm) linking one part of the chip (the part near the ac jack) to the end of the resistor group (opposite the ac jack in the laptop) Also try another combinations in that area However one resistor pin should shurelly come to the part of the chip near the ac jack. Attempt to power back again.  If you do all those things all right the darn thing should work. Make shure that all the components I specified are there in their specific location.
If this doesn't work you may have an error either on another chip that sends the Power On signal (the circuit is burnt, broken, chip damaged etc) or in the charger. Only 2 pins have the 19.5 - 20 Vcc tension, the third is used for Power On signal.

Hope this will help. However Do not attempt to do this changes in the power circuit unless you have some knoledges of electronics and the proper tools for the job.