Yep, I ran into a situation EXACTLY like that at work maybe a month ago. Replaced the motherboard and it worked great. If the thing's under warranty, call Dell and get a replacement, otherwise you can see what it'd run at http://www.parts-people.com to do it yourself.
I'm working on a computer for someone, a Dell Inspiron 1520 notebook, about 2 years old. The computer does not respond to the power button at all. Pressing the battery meter button on the battery itself indicates that it has charge. When I connect the AC adapter to just the wall, the green light on the adapter turns on like it should, but the instant I connect it to the laptop, it turns off with a click. I have to unplug both ends and reconnect it to the wall to get the light to turn back on. I have tried other outlets, and it will not boot regardless of whether the AC adapter is connected.
Sounds to me that something is shorting out on the motherboard. Is this a "replace the whole motherboard" situation, or could something be repaired? I'm not experienced at all in soldering, but might find someone who could. Could it be the DC jack itself? Thanks for any help.