Cessna 172 Flight Instructor reported severe engine roughness during cruise and made a precautionary landing.

Narrative

I was conducting an instructional flight with a student pilot in a Cessna 172 originating from ZZZ1. The flight was conducted under Part 61. We departed ZZZ1 at XA:33 and proceeded approximately 15 NM northwest to our company's practice area.The student performed slow flight; one power-off stall; and one power-on stall. I then simulated an engine failure by reducing the throttle to idle at approximately 4;500 feet MSL. The student identified the simulated failure and responded appropriately. I subsequently escalated the scenario to a simulated engine fire. The student immediately rolled into a 45-degree bank and pitched for 120 KIAS. Once the emergency descent was established; I allowed it to continue until 3;000 feet MSL; at which point I instructed the student to recover; and he complied.We discussed returning to ZZZ1 for landing practice; but elected instead to set up for S-Turns. The student descended from 3;000 to 2;000 feet MSL over roughly two minutes to prepare for the maneuver. Upon reaching 2;000 feet MSL; approximately half a mile prior to crossing perpendicular to a road; we experienced severe engine roughness; comparable to cinderblocks in a washing machine.I took control of the aircraft and immediately decided to divert and land at ZZZ; approximately 3 NM to the north-northeast; without prior permission from the airport owner/operator. I requested priority handling with ZZZ Approach. Approach informed me they would relay the information to the airport's owner.After landing safely at ZZZ; I remained in contact with ZZZ Approach and informed them of our status. They advised that ZZZ was fully aware of our landing and had no issues with it. They also asked if we required assistance; which we declined.On the ground; I contacted the ZZZ chief pilot and airport manager for ZZZ. He was extremely helpful; expressed relief that we were safe; and stated he had no concerns regarding our landing at his airport.