Are you open to using jQuery, it would make things a bit simpler.
I'll try to show you what VIPStephan means using a jQuery example:
//we're now inside the callback
//perform actions with data returned from search form
//this is where you can put your event for your button
Your button will have to be on the page: "url-to-my-search.html"
I just ran into this issue myself the other day, and this is how I resolved it.