A Flight Instructor reported the student lost control of the aircraft during the landing flare and would not relinquish controls in a timely manner resulting in landing on the grass next to the runway.

Narrative

While giving an Aircraft Checkout to my student and after completing a proficiency check on basic maneuvers and stall recoveries I made the decision that he was satisfactory. At that point I instructed my student to fly us back over to ZZZ so that I could evaluate his traffic pattern and proficiency on landings. He performed two landings; each of which included too high of round-outs and flares; which resulted in becoming too slow; too high above the runway surface and ending in firmer than ideal landings. On the third landing; I took control of the aircraft on the downwind leg of the traffic pattern to demonstrate a proper landing and in the hope of showing him the correct sight picture and distance above the runway surface for the round-out and flare phases of landing. After a touch and go on that third landing; I gave control of the airplane back to my student on the downwind leg of the traffic pattern and once again he rounded-out and flared too high above the runway surface; resulting in another hard landing. We only had time for one more landing. I had already made the decision in my head that the result of today's aircraft checkout; solely based on landing proficiency; was going to be unsatisfactory; but still wanted to use this final opportunity to help him work closer to properly executed landings. On our final approach to landing; my student continued his same tendency of rounding-out too early. This final round-out was actually higher than all previous attempts. Our airspeed was now around 45 KIAS; Vref is 54 KIAS. During the high round-out; I instructed my student to relieve his elevator back pressure to allow the nose to come down towards the runway so that we could get back on our Vref airspeed and then continue to round-out closer to the runway surface. Instead; he pulled back further on the control yoke and I then instructed him to go-around. He applied full throttle and full elevator back pressure which together did not begin a climb as the aircraft was approaching the critical angle of attack. The aircraft's stall warning system was warning us and I decided I had to take control of the aircraft.During my pre-flight briefing I explained positive exchange of controls and that if I say; 'My controls;' he needs to release all control of the aircraft immediately. This is a procedure that he informed me he was familiar with and performed satisfactorily previously in this very flight when I demonstrated a prior landing. However; in this critical instance; my student seemed to freeze on the controls with all his strength and after repeating 'My controls' I realized I was now in a fight for control of the aircraft. His uncoordinated inputs of the flight controls now had the aircraft on the verge of stall about 50 feet above the runway surface and the aircraft was traveling at a 45 degree angle to left off of the runway. After several critical seconds of fight for the controls; he finally let go of all control pressures; but we had continued to sink too close to the ground and were now above the grass/dirt to the left of the runway by the time I had full control of the aircraft. There was no time or altitude left to attempt a landing on the runway; so I was left with no choice but to touch down on the grass/dirt to the left of the runway. Touchdown was firm on the main wheels and the ground left of the runway dips down and then up again in between the runway and the parallel taxiway. To avoid going nose down into this dip and also prevent a propeller strike on the ground; I applied full elevator back pressure. As I began taxiing the aircraft through the grass/dirt into the upward inclined portion of the ground prior to the taxiway; the tail of the aircraft struck the ground. I was able to land and taxi in between runway lights and taxiway lights so there was no damage to the airport surface. Upon reaching the taxiway; I kept control of the aircraft as I taxied us to the ramp to inspect the aircraft for damage. Upon reaching the ramp and shutting down the aircraft; we found that there was damage to the bottom of the aircraft's tail. A plastic circular piece that was once attached underneath the tail in case of a tail strike was broken off and there was some cracked and warped metal around the housing for the broken off plastic circular piece. During debrief with my student; my first goal was to ensure he was physically uninjured; which he confirmed to me he was and that the incident did not result in any physical injuries. At that point; he did however express he acknowledgment of fault in the incident and a 'bruised ego' due to his uncoordination and lack of proficiency when it came to his execution of critical landing phases of the flight. After finishing up with him and debriefing him on his lack of cooperation with my instruction and failure to properly respond to my commands for the exchange of aircraft controls; I returned to the aircraft for further inspection. The only other damage I found (in addition to the bottom of the aircraft's tail structure) was to the rear corner of the pilot's side aileron. The corner of this aileron is slightly bent; warped and scraped up.