We need some “Nixon to China” or “Sadat to Jerusalem” level outside-the-box thinking on this. Maybe Obama should personally challenge Alex Jones, Wayne LaPierre, and Rush Limbaugh to a public debate, individually or all at once. All at once would be more fun, but the result would be the same in any case: Nutcase idiots humiliated for their refusal to participate or, better yet, exposed as jibbering frauds in front of a national audience.
The “adult” GOP would object, of course, on the grounds that these people are insignificant gadflies. That lie could be quickly exposed with poll results and concrete examples of their influence.