MKC Controller described a conflict event between an IFR departure and a local photo mission aircraft when missed approach instructions were misunderstood between controllers.

Narrative

MKC/MCI were in a South flow. Runway 19 was the main runway. The Cessna was in contact with Departure East maneuvering about 5miles SW to SE at 4500 on a photo mission. MCI instructed MKC to stop jets at 4000. Satellite South called and gave me missed approach instructions for Aircraft X on a practice ILS to Runway 19. The instructions were 'give Aircraft X standard jet departure and 318.1'. The standard jet departure is turn right heading 230 and maintain 5000. At that time the Cessna was 5 miles SSE and moving east so I didn't think to much of it. As Aircraft X started to execute his climb out I observed the Cessna had turned back to the West and was closing on the T38. I issued traffic and the pilot 'rogered' them but did not call the traffic in sight. I issued the traffic again and the pilot stated that he was level at 5000. The T38 climbed so fast that the Mode C went from 3500 to 5000 in one sweep so I don't know how close the airplanes actually were to each other. I would estimate 1 mile and 500 feet. The Approach Controller should have been more specific in his instructions and I should have questioned him about the altitude restriction.