ATCS reports aircraft crosses runway threshold and lands while preceding aircraft is still on runway for touch and go.

Narrative

Aircraft 1 calls reporting inbound 20 miles southeast. LC requested an ident. Ident received by an aircraft southeast at an altitude of 3000 ft MSL in the airspace of another airport. LC then tells Aircraft 1 to contact Falcon Tower. After a few minutes the Aircraft 1 comes back on frequency saying he is inbound 6 SE. LC issues a left base entry and to report a 3 mile left base. Aircraft 1 acknowledges. Aircraft 1 is then sequenced to follow an Archer on one mile final. Aircraft 1 says he has traffic insight and will follow. Aircraft 1 is then sequenced and cleared to land. An aircraft on the upwind makes an early right turn for a downwind departure making him a factor for traffic currently on the downwind. LC issues traffic and makes the necessary measures to avoid an imminent situation. By the time LC gets back to the runway Aircraft 1 has crossed the departure (arrival) threshold while his traffic to follow is still on departure roll for touch and go; separation lost at threshold. Aircraft 1 exits the runway and begins taxi without contacting ground. Aircraft 1 should have been handled more carefully; especially after seeing the amount of errors the pilot had made. LC controller should have been scanning over everything rather than focusing on a single problem. Multi tasking and scanning exercises would be beneficial.