If you haven't already, though I suspect you probably have, disconnect everything from the motherboard unnecessary to boot from the motherboard, including the LCD, keyboard, wireless card, etc. You just want the power button, battery, and one stick of memory. Try turning it on with an external monitor attached if possible (if not, keep the LEDs connected). See if it behaves the same way. If it works now, reattach things one piece at a time till it's working. Shorted components can cause weird behaviour like this.
You could also try to re-flash the BIOS. First you'll need to find it - look for a memory chip, usually in a SOIC8 package, and it's usually somewhere near the CMOS battery connections. Buy yourself a buspirate if you don't have one and and use flashrom to write the bios file onto the chip.