A Tower Local Controller providing OJT reported they did not notice their trainee cleared an aircraft to line up and wait from an intersection on a runway when another aircraft taking off from full length overflew them by less than 200 feet.

Narrative

During a training session on Local 1; my trainee had cleared Aircraft X for takeoff on Runway XXR full length. Departure called requesting the transfer on an Aicraft that had departed IFR and was previously transferred. The IFR Aircraft was eastbound flying towards ZZZ finals and this took immediate priority in my opinion which is what I was focused on. My trainee decided to issue an instruction to Aircraft Y 'Runway XXR; Line Up and Wait'. I ignored this instruction and went to the IFR Aircraft to transfer control to Departure again. From where I was standing I could not see the strip for Aircraft Y. I did not know that Aircraft Y was at intersection 1. I turned back to the Runway in time to see Aircraft X airborne; over flying Aircraft Y; who had taxied onto Runway XXR at [intersection] 1; by approximately 150-200AGL. I was focused on several things that the trainee had going on. He had just turned 2 VFR Aircraft right at each other to the southeast. He didn't get a readback on the IFR Aircraft when he transferred control; which prompted me to have to call Departure and let them know the pilot needed to read back their instructions. When Departure called back; it was clearly a situation that needed to be fixed right away. When the trainee told Aircraft Y to Line Up and Wait; he didn't use 'at [intersection] 1' which we later found out; he didn't read the strip and also didn't know Aircraft Y was at [intersection] 1. From where we all have to stand while training local 1; it makes it very difficult to read the strips in the bay; as the bay is located between local 1 and ground control.