Light aircraft flight instructor reported a near miss with another light aircraft in the pattern at MKL airport.

Narrative

I was acting as an instructor for a part 61 training flight. My student and I were staying in the traffic pattern doing landing practice in Aircraft X. Also in the pattern with us was a student pilot solo also in Aircraft Y. This was a particularly busy day and there were more arrivals than average when we were in the pattern. At the time of the near miss; I was on the left downwind for runway 20 and the student pilot was on the right downwind for the same runway. We made a left midfield downwind call to ATC; and was cleared for the option number two behind an aircraft that was on final. We responded confirming that we were cleared for the option number two and that the traffic on final was in sight. Shortly after the student pilot on right downwind made their midfield call and were instructed to follow Aircraft X turning base for 20 (me and my student). The student pilot then reported traffic in sight and ATC cleared them to land number three. My student and I continued flying a normal traffic pattern and were in final when the student pilot came on the radio and reported a 1.5 mile final. This alarmed me since our GPS was indicating we were 1.5 miles from the airport. I glanced at foreflight on my device with ADSB-in capability through a sentry and saw a flashing red traffic alert. I looked left; right; and above me and about 50-70 feet above me and slightly to the right I saw Aircraft Y above me. I immediately took the controls from my student and applied full power; dropped the nose slightly to gain airspeed and banked as banked to the left as aggressively as I felt was appropriate for our airspeed and configuration. I notified ATC we were going around and breaking off to the left. We re-entered left traffic for 20 and completed the flight with no further incidents. I think this incident was caused by one of two things: 1) The student pilot mistook the aircraft on final for us (on downwind) and followed them in. This would explain why our base to final spacing was almost identical. Or; 2) The student pilot did have us in sight but got fixated on flying the airplane and lost awareness of where we were.Factors that I believe contributed to this incident were two main things. I believe that the student pilot was likely not ready to fly an airplane solo and also maintain situational awareness to the level that was required on a busy day with a lot of arrivals. I also believe that the lack of radar facilities in the tower could have prevented incidents like this from happening. If the tower was equipped with radar; they would have seen myself and the student pilot converging and would have been able to help resolve issue before I needed to take evasive action.