This is what I would do noting that I have replaced horn wires and repaired in length and straitened horn roads in my '32 and '40.
The challenges you have are that there is no horn wire stub to connect to and that the horn road has been completely cut off.
1) To replace the horn "ground" wire you
must take the horn assembly apart by bending the four tabs up.
"On the underside of the button module there are four tabs that are bent over. These tabs are what hold in the bezel that holds the button in place
Using a shop knife I pry the tabs up to the point where you can get a needle-nose pliers in to straighten them up to vertical."
2) With a horn wire repair kit you will use the parts and ferrule included. The top of the wire must be crimped and soldered.
3) Slip the tabs back into the module and carefully bend them over so the assembly is tight. If you break them, clean and JB Weld epoxy the stubs in place.
4) Now your reassembled module will have a wire tail sticking from it.
5) The wire tail is now to be threaded down the steering shaft and out a hole just above where you will bond an insulated copper sleeve onto the shaft.
6) Before the copper sleeve is installed solder a short pigtail on 12 to 16 gauge wire to it. You will later crimp or solder the wire from the horn button to it.
7) Cut an oval hole lengthwise that will be covered by the Jeep contact switch ,
https://www.kaiserwillys.com/products/steering-column-horn-contact-brush-fits-41-49-mb-gpw-cj-2a/
8) Slip the steering column pipe over the shaft now and mount the steering wheel
9) Now you need to solder or crimp and then insulate the short pigtail from the Cu sleeve to the horn wire pigtail. Keep the excess wire to a minimum
10) Push a wire zip tie in the oval hole, pull the wire from the horn button so it is snug and tighten the zip tie. Add another zip tie through the oval hole and snug up the excess wire so it's not going to rub on the steering column.
11) Now mount the Jeep contact brush.
Regarding replacing the horn rod, to me it's not needed and would be difficult to do since you have no stub to attach to. The original is swaged over. You would have to solder a new rod stub while keeping it perpendicular to the horn module and making sure that your solder fix doesn't interfere with the ID of the steering shaft, holding it up. If not, your horn module will be canted off to the side and rub and turn with the steering wheel. Even if you are able to add a stub, the only thing holding it in would be friction between it and the inside of the shaft, You still have the extra wire to deal with as I mention above.
