Inspiron 9100 power problems

My inspiron 9100 is having problems. It just shut off one night. i thought it had overheated so i turned it back on.. then it shut off again. Now when i push the power button, the keyboard lights and the power light comes on, but nothing happens, they stay on for a few minutes then they just go off.... however one in like 30 times it will actually come on... so to make sure it wasnt a software problem i reformatted it, but its still the same thing. the dell support instant message people are idiots, they tell me to do all this stuff that i know wont work, they dont seem to grasp my problem... so anyways ive narrowed it down to some sort of hardware problem. because sometimes it will turn on when i change something, like take the battery out, or unplug it, or plug it in. so it seems somwhere between the batter, the power cord, the power button, and the motherboard... somewhere the power isnt gettin where it needs to go...

any suggestions?