PA-28 Instructor pilot reported an engine malfunction during climb on a training flight. The Instructor took control and returned to the non-towered airport and landed safely.

Narrative

My student and I were conducting a 141 training stage check when the powerplant suddenly changed tone and began to shake violently. At the time we were at approximately 700 AGL on a climb out from a simulated power-off landing which was recovered from around 500 AGL. Immediately after the engine roughness started I assumed the controls and made a turn towards ZZZ (only eight miles away) and performed the engine roughness checklist from memory then instructed the student to use her checklist and ensure I had not missed any action items. Once this was completed; it was apparent we would not be able to maintain altitude so we continued on full power and were able to maintain 600-700 AGL at 80 Kn. We then proceeded into the airport and changed the squawk code to XXXX to signal for help in case of a complete engine failure. We were able to clear out the final approach corridor for runway five after [saying we were inbound] aircraft on the CTAF frequency. Next; we maintained the best possible altitude with full power until we were within gliding distance of the runway and proceeded to land safely.We later found out from the maintenance team that this was a stuck valve on piston three that was likely a result of a previous crew improperly leaning the mixture.