From a couple minute web search, it appears to me you do not need to run an additional wire for the trailer brakes. In addition to purchasing the controller (i suggest Tekonsha P2) and plugging it in under the dash, you may have to add a fuse or relay under the hood to complete the circuit. Very simple to do. Its common that vehicles equiped with the tow packages have all the wires included in the harness, but the factory may not have completed the circuit by insalling fuse/relay in the underhood fuse box.
etrailer has lots of videos for isntallation that may be helpful.
I suggest this controller and easy installation cable.
I also have this to test the 7 way connection at the back of my truck. It will tell you right away if the 12V constant is active as well as brake controlls, indcator lights and running lights.