Several possibilities here. The battery could have been damaged so it is not providing the required power to the board. You could try disconnecting the battery and plugging in your AC power adapter alone and try running the computer.
The high temperatures inside the car may have warped or flexed the BGA solder connections underneath the Intel CPU or the soldered on memory module ICs just enough to cause them to not be solidly soldered to the motherboard - even causing what’s called “micro-fractures”. The process of “reflowing” may fix this problem, but without knowing for sure if that is what happened, it could make things worse.
If it were me, I’d use a multimeter and try to measure power at key points on the board to see if it appears to be getting proper power. Measure the voltage coming out of the battery and at the battery connector when plugged in and on.
You may be able to find schematics for the board if you Google the motherboard model number along with “schematics boardview”. That would definitely be helpful. If you had a modern POST card with mini-PCIe connectivity, you could check and see if the processor or RAM is functional.
Finally, should all else fail, you would have to replace the motherboard since all of the major components are soldered to it.