Asking for advice on soldering 2 thin copper wires to a motherboard in a vehicle ignition to get spark to start the engine.

 

The Ignition-Switch in a Car may be required to carry very High-Currents of over ~40-Amps.
I can't tell by your pictures exactly what You are working on, or what may have broken or melted.

But taking a wild guess ........
You probably can't "patch-it-up" and have it last more than a week.
The repair / replacement will very likely require a proper Wiring-Schematic, and proper testing,
and/or, an OEM, or equivalent aftermarket replacement Switch-Assembly.
( A Junk-Yard part might turn-out to be the cheapest way out )
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Why are you assuming it enables the spark ? First find out what the printed circuit board is for. Seems like a keyfob proximity presence detection, can be a buzzer feature, an anti-theft alarm, a key -inserted sensor, or something else.
If you are skilled in soldering and doing a straight repair of broken wires, go ahead with the battery disconnected. If your plan is modifying or inventing something, better abstain and dedicate to learn what is the circuitry involved first.