Damaged trailer wiring harness

Those are scotch loks, aka junk. A butt connector crimped with the proper crimping tool--NOT a pair of vise grip pliers--and covered in shrink tubing will last a long time, and is less prone to failure than a solder connection in this kind of application. As someone else noted, solder is an excellent electrical connection, but a poor mechanical one. That said, I'd go with the advice to simply replace the entire cord with one that has a molded on connector. Do it once, be done with it.