That's exactly what the problem was - that little wire was melted. I was going to remove the insulated washer between the two copper pieces to bypass that little wire, but right before I tested it, my mom texted and said I should have my dad look at it first. Today I brought it over and he soldered a car fuse to it connected by short pieces of wire. It doesn't look great, but we just stuffed it back into the dash and now the socket has power! :)