Okay, here we go........
You need to get up under the trailer.
The red wires coming from your trailer brake are should be twisted together at some point into one wire which travels to the trailer plug. The two brown wires are grounds and they should be bolted, or screwed, to the trailer's frame somewhere. Make sure those things are correct. When you trace the place where the red wires come together, they will be, at that point, connected onto a single wire that goes to your trailer plug. Make a note of the color of that wire. That SHOULD be your blue wire that you are looking for, however, it does not HAVE to be. ANY color wilre will work, so long as it is a 12 gauge wire. So, if it is NOT blue, don't worry, just look at the wire, note it's color, and check the wording on it to make sure it is a 12 gauge wire. Now, that wire goes into the trailer plug at the pole marked "S." Make sure that it does. If there is no wire connected from the TWO reds to the "S" pole on the plug, you will have to add that wire. Again, any color is fine, just make sure it is a 12 gauge wire.
Now, go up to the brake box in the
truck, and trace your blue wire there. You said it is spliced into a black wire, you think. That is fine (again, so long as it is a 12 gauge), but you need to make sure, and then you need to follow it back to the truck plug. That wire (the color does not matter, so long as it IS the wire from the brake box) should go into the "S" pole on the truck plug, too.
That wire from your brake box feeds juice to the magnets that are in the trailer brake assembly. When that juice hits the magnets, they jam up against the sides of the brake drum and that causes the brake shoes to engage.
Now, here's something that is critically important. If the reason you are working on this is because you've previously pulled this trailer and suddenly the brakes are not working, the magnets themselves may no longer be any good. You MUST TAKE IT TO BE SERVICED!! The brakes will NOT work,
at all, if the magnets are worn out. However, if you have never previously hooked to this trailer, then you MUST trace each pole coming out of the
truck's plug (or use a tester light) to make sure that that is wired correctly. Here is how it should be connected to the plug:
Trailer brake box wire: "S" pole
Left turn signal: "LT" pole
Right turn signal: "RT" pole
Ground: "G" pole
Auxillary (for trailer interior lights--optional): "A"
Trailer Marker lights: "TM" pole
Do this even if someone has told you that the truck is wired correctly. A case in point, we just bought a new truck, hooked it up to our horse trailer, and upon hitting the right turn signal, the left TRAILER signal would flash, and upon hitting the left turn signal, the right TRAILER signal would flash. Someone had gotten the wires for the signal lights connected to the wrong poles.

Obviously, if they were pulling a trailer, the trailer lights ALSO would have had to be wired backwards like that too. Unless they simply never noticed that their signal lights didn't match.........
Now, for the trailer plug, just do what I described earlier, make sure that the two red wires come together and that both are connected to a wire that runs into the "S" pole in the trailer plug, and that the two brown wires are attached to the frame of the trailer. If you have interior lights in the trailer, then the wire for them will run into the "A" pole. You say that the black wire on the TRAILER is the "running lights." If so, then that wire should be connected to the "TM" pole in the trailer plug.
Remember, it literally does not matter what the color of the wires are, but it does make things a lot less confusing if everyone uses the same colors.

And for the individual, it is a lot less confusing if the same color wire is used for the same pole connection on each plug (so if you have to run a wire from the "S" pole on the trailer plug back to the two red magnet wires, use a black one to match the one that the trailer box wire is spliced onto, or use a blue one if that is easier to remember, since you already say you have a black wire for the trailer marker lights).
If you have any questions, let me know.
