First thing you should do is determine if the trailer uses a ground through the connector or the trailer tongue. This can cause quite a bit of confusion and throw you off on your troubleshooting.
This link really covers just about every application out there.
http://www.etrailer.com/faq-wiring.aspxI highly recommend using sealed splices instead of the cheap crimp type, if you have to use the crimp type put a piece of heat shrink over it to seal the splice.
Wrap the spliced area with some electrical tape and finish using a section of wire loom tied off with wire ties. That's how I always fix my trailer wires and it makes it look professional too.