The little contact plunger pin and spring on the original wheel is the horn switch that completes the ground on the control side of the relay. put it back in the hole with the wheel off and use a piece of wire to ground the end something while pushing to contact. It should fire up the horn if everything is working except at the wheel.
If the horn works, then you'll have to mod up something to make the new wheels horn button cap work.
What I saw on their page looked like their horn cap has a switch in it and a wire from the switch wrapped around a bolt to ground it. Make sure that didn't come loose. You would have to rig up the original or something else in the plunger hole so it is continuous contact so the only switch is the new wheels switch.