Good videos. The second shows the sort of 'amateur' repair you would have to attempt. The biggest problem is that you would be trying to solder aluminium wire. This requires special solder and flux, and good technique since the oxide layer which prevents solder adhesion, reforms as soon as air gets to the wire. I don't think replacing the wire with a copper one will work because the pad on the circuit looks like aluminium to me.
Except that: it might be possible to trace the circuit back from the aluminium pad to the first component connected to it, and solder a copper wire to it, with ordinary solder. Then solder the other end of the wire to the (gold plated?) pad the original wire came from. You need to know what you're doing though...
Also, of course you are working on a safety critical component. We, most of us, do that when we change the brake pads etc etc, but they're harder to break! Unless you feel confident about tackling this it might be better to have it repaired professionally, in the manner of the last video.
Sorry, can't help with the lights.