Hello Brandon, i think these new column switches come with two short wires already connected. The column switch is just used as another connection point, from the 12 way connector under the dash, the loom travels up the column, the horn wires terminating in the brown, ground/earth terminal and the black/yellow/pink terminal. The short wire from the brown terminal is connected to one side of the contact plate, and the other terminal of the contact plate is the switched earth returning to the black/yellow/pink terminal, which then travels back down the column, eventually terminating at the left hand side horn. There is a permanent +ve on the other side of the horn. If you short these two wires together, ie the brown and black, or brown and red wire in your case, the horn should sound, but don't forget the ignition needs to be on
The colours of the short flying leads from the switch probably vary depending on the switch manufacturer
The fact that a separate horn switch has previously been fitted, probably proves that your contact plate failed at some time, although they must be quite simple to repair, clean or adjust
Good luck!
