I doubt WGN will try to get a national sport deal as how TBS did when the Braves left they went after a MLB package for Sunday during the day and post season games. As for WGN its hard TNT has NBA so its hard to say what sport they would get.
Now that all four major network ownership groups have dedicated cable sports networks (although CBS Sports Network is a pretty half-assed attempt at one), it's hard to imagine WGN America coming up with rights to anything compelling to great numbers of viewers. Can't imagine MLB or NFL wanting to add outlets -- and chances are, the price the NFL would ask is too rich for WGN's blood. The NHL seems happy with its NBC/NBCSN deal, and the NBA has ESPN and Turner. I suppose they could find a boxing or MMA promotion to hook up with, or some second- or third-tier college conference that has no appeal to the big boys, but what's the point? They could just draw on their library of old sitcoms and dramas and grind out an unexciting, uninspiring profit, as there are plenty of non-sports fans out there, as well as plenty of people who could watch their favorite old shows ever night until the end of time.