Tower Controller providing OJT described a 'departure prior to an arrival' loss of separation event when failing to intervene early enough as the student was not keeping up with the traffic.

Narrative

I was providing OJT on LC1 (LC2 was combined; which is normal); with moderate traffic and the developmental was struggling. I was pushing the developmental keep up with the traffic. However; there were a number of tasks that needed to be completed in a timely manner (extending one aircraft's downwind; sequencing; etc.); but I was trying to allow the developmental an opportunity to demonstrate his ability. The developmental cleared Aircraft X; a PA46 for take off on Runway 29R with Aircraft Y (C340) just inside a 3 mile final. At this point; I sensed an overload was about to occur; and I took over; clearing Aircraft Y to land and advising him of the PA46 departing. C340 responded that he had the departing 'traffic in sight' and I turned my attention to other matters. However; at this point I became distracted from 29R; and was scanning to find the position of Aircraft Z exiting onto D Taxiway; South of 29L. Multiple other aircraft were calling inbound as well. I was concerned about Aircraft Z (student pilot; solo) because she was very challenging to work; and her radio ability/technique sounded quite questionable at times. Just a couple minutes earlier; on the second attempt; Aircraft Z correctly read back that she was cleared to land Runway 29L. However; as the OJTI; I observed the aircraft roll out on final for 29R. The developmental did not see this; and I took over as the instructor; instructing Aircraft Z to turn left to line up/land on 29L as assigned. The pilot apologized; corrected and complied. However; after Aircraft Z did not acknowledge exiting instructions on the developmental's first transmission; I was concerned with what she might do next. Aircraft Z did read back instructions on the second transmission and exited 29L. As a result of all of the factors involved; I believe too many things needed to be accomplished simultaneously and I was not watching runway separation closely enough on Runway 29R. Though I saw Aircraft X turning the corner onto 29R; I speculate that Aircraft X stopped on the runway before starting take off roll. Next; I observed Aircraft X upwind near A4; with Aircraft Y touching down on the runway (thus I believe I had 3000' and airborne; but not the required 4500'). There were no complaints made by either pilot; and Aircraft Y taxied to parking without incident. Recommendation; this event was frustrating because my initial plan; had I been working the position alone; was to have Aircraft Y change to Runway 29L for landing as soon as he came on frequency (thus balancing traffic flow between the two runways; managing my workload; and allowing plenty of time for multiple 29R departures). However; the developmental is so far along in his training that he needs to demonstrate he can dig himself out of a hole; and thus I was trying to maximize this training opportunity by allowing him to work himself out of his situation. This was a strong reminder for me as an OJTI of the importance of balancing providing good training; while ensuring I preserve safety and service by not allowing a situation to exceed my capabilities. I believe that after Aircraft Y reported 'traffic in sight;' I wasn't watching that situation as closely as I should have; and may have assumed somewhat that Aircraft Y would then adjust his approach. This event is also a good example of different standards/expectations between pilots and controllers. In this case; Aircraft Y's concern might have just been keeping Aircraft X in sight and ensuring he was satisfied he would not collide with the other aircraft. But even if the separation was acceptable to the pilot; the higher standard of 4500' and airborne must be met in this case.