I think they call that the button assembly - that is NOT a cheap part. Retail price is something north of $700 for just the horn/pad assembly, don't think they just sell the wiring portion you need separately. Might be able get one from Ebay for a lot cheaper.
Probably have better luck trying to fix the connector to the horn yourself (try and resolder the pins to the connector), or try and pickup a part from a local salvage yard. Any 9th gen steering wheel should fit - just make the color. I don't remember a wire running to the horn though, that sounds like something with the clock spring or spiral cable assembly - ie, the airbag.
From what I remember, the horn connector should be right above the center nut - fixed to the steering wheel portion. Nothing to the center pad except the wiring for the airbag.
I'm assuming you double checked the horn relay / verified 12V directly to the horn - before you went through the process of pulling the steering wheel apart.