If you still have the failed laptop, I can think of two things to try.
It is likely to be one of two things, either graphics chip / Chipset failure, or failed switching circuit on power up sequence.
1) Failed chip onboard;
Dismantle laptop and take out motherboard. Strip everything off it you can (inc. CPU, memory modules etc).
Wrap the motherboard up in tinfoil ensuring to cover everthing especially plastic areas. Cut around the suspect chip (graphics and or chip set controller) and remove tinfoil over chip, so that only that chip is exposed. Create 4 tin foil balls and get a baking tray. place balls on tray and place wrapped up motherboard on top of balls. This aids airflow.
Put in oven and turn on and let it heat slowly fan unassisted at 180deg. Once at 180 cook board for about 8-10 minutes then turn off oven, take out board and let it cool down still in tin foil. This will re-flow the failed chip. Once cooled, re-assemble everything and test.
If it works, backup everything you need off it, and sell it quick. It will likely last for 3-6 months before failing again. If you can afford the cost, you could take it to a specialist and get the suspect chip reballed and reflowed instead but this is usually an expensive option.
2) Failed power circuit.
This is a little more difficult, but if you are good with electronics and a soldering iron, trace from the power in socket all the mosfets in circuit and test each one (look up its specification online) - its likely you will find one thats failed. Buy and replace failed one. Typical symptoms of a failed mosfet are things like power light shows and charges battery, but pressing power on, power light flicks for an instant, then machine shuts off, or power adapter wont charge battery, but battery can be charged in another machine no problem, and that battery will power up your machine fine, or laptop will power up on mains, but wont charge battery, yet battery charges ok in another machine.
Jason.