I bought a bigger soldering gun ($15, Harbor Freight) and did manage to solder the ground successfully. I couldn't solder the other two as I couldn't hold them, only had the two hands.

So, I stripped them down. I planned on coating them with solder and trimming them to fit, as you suggested. However, when the stranded wire came out of the insulation it was quite compactly twisted together. I was able to easily fit them onto the prong and screw it securely. I fit everything back into plug body, coated the entire mess with liquid insulation several times.

Thanks for the suggestions. They gave me hope as I bumbled my way to something which seems to have worked.