FWIW, a true trailer wire has different gauge wires in it. The power and ground wire are heavier gauge. Besides color, a web search would lay out what the wire gauges should be. I would try to match them to industry standards.
The two power lines are 10 ga., the brake is 12 ga., and the lights
(turn, backup and running) are 14 ga.
You've already purchased but for about $27 you can get the plug, a length of wire and a pre-wired junction box which would make hooking everything up easy.
I'm thinking of one of these because I may add an electric tongue jack and it would be an easy way to connect the electrics.