D10 Controller providing OJT described a near wrong airport landing event involving a C550 cleared for a visual approach to DAL; the aircraft flew toward DFW and failed to contact DAL tower as instructed.

Narrative

Training was being conducted where I; the instructor; had just finished working the position for approximately 15 minutes due to volume and complexity; the Developmental had just resumed the position and maybe only talked to 1 or 2 planes when the event began to happen. The Developmental asked Aircraft X if he had DAL in sight; the pilot responded affirmative and the Developmental cleared Aircraft X for the visual approach to Runway 13L at DAL field and switched him to DAL Tower; when the Developmental cleared Aircraft X for the approach the aircraft was still on a 4 mile left base with about 8 flying miles to DAL. The Developmental noticed right away that the C550 was not turning toward DAL in a timely manner and called DAL Tower immediately and asked them to make sure that the C550 was turning southeast bound to the airport. The Tower tried to call the plane but he was not on their frequency; now he was quickly approaching DFW's short final airspace with a MD80 at 2;000 on a 4 mile final. I called DFW Tower and pointed out the C550 to them and told them that at the time nobody was talking to him. They accepted the point out before any separation was lost and right about this time DAL Tower called us and told us that they were talking to him and turning him back to the east. I told them to expedite because of the traffic right off his right side. He turned east and landed at DAL without any further incident. DAL and DFW are so close that if any aircraft overshoots final at DAL; or misses a turn from us; it quickly impacts DFW's traffic.