ZID Controller described a potential conflict when one aircraft experienced a 'stuck microphone ' and was unable to communicate with conflicting traffic; which had entered the airspace without coordination and/or hand off.

Narrative

Aircraft X was an aircraft climbing on a vector. After acknowledging a clearance back to the FWA VORTAC; Aircraft X unknowingly had their mic stick. At that same approximate time; a limited data block was identified approaching the boundary at nearly the same altitude that Aircraft X was climbing through. I immediately called for the WolfLake Sector at Chicago Center. The controller there thought he had transferred RADAR control of Aircraft Y but had not. I told him to turn Aircraft Y; which was the limited data block; 25 degrees right because that aircraft was in conflict with Aircraft X. The controller at Chicago center said that he had given Aircraft Y the frequency change to our sector. With the stuck mic; we were unable to talk to either aircraft. Fortunately Aircraft Y out climbed Aircraft X by a substantial margin and no separation was lost though the situation was uncontrolled. After a couple minutes; Aircraft X checked back on the frequency and was notified of its stuck mic. Aircraft Y then was able to check on to the frequency as well. Aircraft Y was never issued the 25 degree right turn that I had requested to the Chicago Center controller. Recommendation; ensure that all aircraft are handed off and that separation is always controlled.