I took the MBA motherboard to the lab at work, and soldered the connector back (or, thin wires from the pads on the motherboard, and it still did not charge or detect the battery.

Either the battery is wasted or the MBA motherboard has a problem somewhere else. But you are correct, the connector is surface mounted. With professional soldering equipment it was not too much of a deal to remove the broken pins from the mainboard and fabricate a fix, but unfortunately, as I said it still does not care about the battery.

Still, the machine runs like a dream from the ac adapter.

I have applied new cooling paste and the machine runs quiet and cool now, at least if you dont play a 3D game or something