the 5.5mV you are reading is nothing, meaning without a doubt you have a faulty horn switch... i doubt it's to hard to fix though, strip it down and clean it up and all should be well
i fitted a nautilus horn other day, you have to use a relay as the stock horn is only a few amps and the nautilus is something like 18amps @ 12v, so without a relay you will cook your horn button
i pulled out stock horn and ran wires from that to the relay coil. then the switched contacts on relay powered up horn with a direct feed from battery via a 20a inline fuse.
nautilus should come with a relay, well mine did :)