If you have a multimeter you should be able to check the output of the power supply. If the output from the power supply is good than it is probably your battery. If I remember correctly the battery had to be installed to power up the laptop even with the power supply.
My latitude 100L making same problems, it works from battery, but not from AC-adapter. Adapter is OK, problem is on the motherboard, I checked with oscilloscope - PWM controller (ISL 6247) on the motherboard give power for short time and for unknown reason, interrupt supply of power. So, I will play a bit again with this laptop, and if it will not work I will probably replace motherboard. Check on ebay, it could cost around 40$.
One thing to check would also be the power cable from the adapter to the computer. If there's a physical break in that line it would result in the same issue, slow battery decline and failure of power. Find another AC adapter and see if it works. If that still fails, then Tyler's advice stands.
