This laptop is, by far, the worst most horrible POS I've ever seen. Worse than the HP DV6000 series laptops. I had the same problems you had and this is what I found out.
The research I did was extensive. Unfortunately it was a long time ago and I forget some of the details, but I'll give you as much information as I have. Apparently this happens when you allow the Lithium Polymer battery to drain too low. There's something on the battery that is linked to the motherboard. If you replace the motherboard, it will do the same thing and you'll damage the new motherboard just by plugging it into the old battery. You need to make sure and replace BOTH the battery AND the motherboard at the same time and don't mix and match them. I don't remember why, but you need to trust me on this.
For me, it didn't seem worth it but my client insisted that she wanted it fixed. I replaced both the battery and the motherboard and it fired right up.
I also heard that the keyboard might be the problem, but I find that hard to believe (though it's not unheard of). In my case, I replaced both the battery and the motherboard and everything went smoothly. But if you want to gamble on the off chance that it's the keyboard, it would certainly be cheaper than replacing both the battery and the motherboard.