C172 Flight Instructor reported engine roughness on final approach in visual conditions. After touchdown; the engine failed on the landing rollout where the aircraft was then towed off the runway.

Narrative

During an engine break in flight with myself and another instructor; I noticed engine roughness while pulling the power back through 1500 RPM while on final approach to runway XX. Below 1500 RPM; the RPMs began free-falling so I put power back in and kept it above 1500 RPM until over the numbers of runway XX. We did not know if we were going to lose the engine; so I erred on the side of caution by keeping power in longer and coming in a little higher than normal. While flaring during the landing; the engine fully quit and the propeller came to a stop while rolling out; at the same time that tower was giving us taxi instructions for the FBO. I informed the tower immediately after realizing; that we had lost our engine and would not be able to exit the runway. The FBO assisted with towing us off the runway. We immediately checked fuel and oil; both were sufficient and could not have caused the failure. After further inspection by an A&P; we discovered that the carburetor mixture idle control screw was missing and presumed to have vibrated free from the carburetor; causing fuel to not make it into the engine at low RPMs.