What's on the back of the adapter?
My guess would be connect the loose wire to one of the tabs from the button (looks like you need to add a quick connect to the wire), and make up a new wire to connect the other tab to the body of the adapter. You can use an ohmmeter/DMM or powered test light (or buzz box) to check if that's the way the button works. When installed, the body of the adapter will be attached to the steering shaft (so grounded through the shaft), and the brush that I assume is on the back of the adapter will contact the slip ring in the steering column. When you push the button, it grounds the slip ring.