Compaq Presario 2800 won't boot from good battery

Hello:

Hopefully someone can help me with this: I can't find the solution to this
problem.

I have a Compaq 2800T that won't turn on when the AC power is unplugged
(battery only) but if I plug in the ac adapter for just enough time to press
the power button and then unplug it , it starts fine and runs off the
battery for over 2 hours. I tried a new battery without success. The Compaq
diagnostics says everything is ok. The only thing i could thing i could
think of was that the battery connectors could be dirty but i have no idea
how to clean those because they are very deep inside of the battery hole.

Any ideas?

Thanks in advance for the help

Marco