I've never swapped an e53 steering wheel/airbag/horn set-up...but if it's like my e38's or most BMW design architecture from the same generational era (e38/e39/e46/e52/e53)...it should be a simple wire that connects to a push-pin that the installer overlooked.

You mentioned in the quote above that the "wires" and electrical connections didn't add up. You never responded if your original steering wheel was heated or not (would mean 4 wires)...nor if the new 3 spoke wheel is heated or not...but if one of them was and the other is not....then that would account for one wire not having anywhere to go. The 2nd wire is probably for the horn (which the installer appears to may have missed)

The other wire you mentioned not having anywhere to go...is probably the issue you have now with the horn not working...the installer probably has overlooked the pin where the wire for the horn connects.

Looks like you need to go back in and plug the horn's wiring into the electrical harness.

EXAMPLE: in the pic below of my e38's 4 spoke steering wheel & single stage airbag...the BLUE wire w/orange protection (condom) coming from the steering column (on the left) and going to the single location on the rear of the airbag/horn component..is the wire for the HORN. The multiple wire harness on the far left of the air bag (purple/white/black/brown wires) is for the multifunction buttons on the front of the steering wheel...and the large yellow connector in the center of the air bag...is the air bag's wiring harness. Again, your X5 has a dual-stage airbag so the wires connections look slightly different...BUT...the same design principle will apply. You should have wires for the airbag, the horn, then multiple wire-harness for the multi-function buttons:

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