On thing I found with some older stock is that the batteries are not up to snuff for some reason. You don't mentioned whether you changed out the battery on the original computer.
Joining the wire and taping it is not an major issue, as Dnvr points out. All it is doing is transmitting the electrical impulse between the sensor and the computer unit.
But I am trying to work out why there was a wire if it was a wireless unit as you imply with your reference to a weak wireless signal (which incidentally, is not surprising...)
As to the second unit, I can't help there except to suggest a new battery (again).
FWIW, I had to figure out how to get a computer to read for Machka on the back of the tandem. Wireless was out because of the distance from the front to the handlebars, and even from the rear to the bars (same issue as you are experiencing). Heck, the wireless on the bars of our touring bikes had about an inch of wriggle room, so nothing was going to happen from the rear.
In the end, I acquired a computer of a brand that also offered an extra long cable from the sensor. I can't remember what brand it was, because it's in storage on the other side of the world. I think it was a Sigma
But it worked flawlessly. I did have to work a bit to get the sensor set up on the seat stay to get close enough to the spoke where the magnet lived.