If I understand you correctly...
With the battery removed from the laptop, and the power supply plugged into the wall, but not into the computer, the power supply has a light that is on. But as soon as you plug it into the computer the light goes out?
1. Check the ac adapter for any kinks etc that might be a hint of damage to the cord... most likely between the little black box and the computer.
2. Even if the cable is in good condition, the power supply "could" be failing internally. Probably only a tech would be able to test its output (both voltage and power capacity).
3. Is there any sign of external damage to the computer, near the point where the cable plugs in, inside that "port", or any where else. (Inside... if the contacts have been damaged, there may be a "short" here.)
4. Do you know anyone else with the same model of computer. If so, try your power supply with their computer and vice-versa.
Short of damage to the computer itself...
My best guess at this point is that the power supply needs replacement.
I have noticed many people are reporting issues with the PSU for this model of laptop.
HTH