Flight Instructor reported while on a training flight they were executing the missed approach and the aircraft did not achieve full RPM resulting in the Flight Instructor taking control of the aircraft and landing.

Narrative

I was instructor pilot in right seat. Private pilot receiving instrument training (learner) was in left seat.Learner flew an excellent LPV approach by hand to DA; peeked momentarily out of the view-limiting device; then initiated a missed approach. Upon adding power; we both noticed that the engine did not achieve full RPM and was running very rough.I immediately assumed control; pulled power to idle; added carburetor heat and full flaps; requested assistance to the tower and said we were landing on the runway; then landed and taxied off without incident. Tower did clear us to land in the few seconds between my assistance request and touchdown; incidentally...although I was landing regardless.We were able to taxi; with ground control clearance; to an FBO where we shut down. I called the FBO at the airport where the learner had rented the airplane (the departure airport was roughly 15 NM away) and asked for assistance. The FBO sent two mechanics in a different airplane to tend to the engine. My learner and I flew back to departure airport in the replacement airplane while the mechanics did their thing.