Flight Instructor with student reported an engine failure while in flight.

Narrative

Aircraft departed ZZZ1 at XA:32 intending for a dual local flight. No abnormalities were detected in preflight or run-up. Oil quantity was at 6 quarts and warm. In cruise flight at 4500 FT; the engine began to experience a partial power loss. Engine started choking and going in and out and we felt overall roughness and saw RPMs dropping then rising then repeating rapidly. Disconnected autopilot and turned towards ZZZ airport to divert. IP (Flight Instructor) ran memory items and SP (Student Pilot) ran engine checklist to troubleshoot for engine failure and followed QRH for partial power loss. Engine was still experiencing the same abnormalities. Called airport traffic and announced we were inbound for a landing. Power was idle at this point and we were fully gliding down to land. Runway XX was in use and there were multiple aircraft in the pattern. Aircraft was lined up too high on runway XX so IP circled to land runway XY and executed a power off 180 to land on the runway. Aircraft touched down at ZZZ at XA:58. The aircraft had enough momentum to taxi us off the runway and onto the ramp. IP called dispatch for help and rescue from ZZZ. We were not asked if we were ok or if we needed anything; and in fact we were asked if the aircraft was damaged first as that was the main concern. My student and I's well being and safety did not seem of concern to management and we were treated as an afterthought. We were told someone was to come get us via plane asap; however; IP discovered this flight was intending to complete an entire dual lesson then come get us at the very end. We were told they were en route and were going to come get us after 'a few touch and goes as to not have to incomplete the lesson'. This was discovered to not be truthful and once pressed on the matter; an individual came to get us via car. Only after contacting a third individual who is an ACP (Assistant Chief Pilot); was there any urgency for our situation and care for IP and SP.