PA28 Flight Instructor with student reported loss of engine power and aircraft vibration in the traffic pattern and continued to landing on the nearest runway.

Narrative

While acting as the Authorized Flight Instructor for a Private Pilot Student; I was conducting pattern work at ZZZ. During the final lap in the traffic pattern; while on the right downwind leg for Runway XXR (with Runways XXR and XXL active); the aircraft experienced a sudden mechanical failure.Approximately midfield on the downwind; a loud bang and hissing noise was heard from the nose section of the aircraft. Immediately following the noise; a significant shimmy developed; resulting in intense vibration throughout the aircraft. Concurrently; the engine RPM dropped by approximately 300-400 RPM; indicating a sudden loss of power.Given the nature of the symptoms; particularly the combination of mechanical noise; vibration; and power loss; I determined that the issue was likely a serious engine malfunction rather than a minor ignition related problem or fuel starvation situation. Evaluating our position and the glide capabilities of the aircraft; I advised ATC that we needed to divert to the nearest available runway Runway XY. Runway XY was within immediate glide distance; and ZZZ Tower granted clearance for landing.We executed a safe landing on Runway XY without further incident. Although the engine continued to operate; it was running abnormally during the taxi back to the FBO.A post-flight inspection was conducted by the on-site A&P mechanic. The inspection revealed that Cylinder #3 had completely failed; with 0 PSI compression; and the remaining cylinders were operating below normal compression levels. This confirmed a significant powerplant malfunction consistent with the symptoms experienced in flight.I am submitting this report in accordance with FAA safety reporting practices to document the event and assist in any necessary follow-up or investigation.