SR22 flight instructor on training flight with student reported NMAC with another aircraft in the traffic pattern.

Narrative

My student and I were returning to ZZZ after a cross country training flight with my student as the pilot flying. I was acting as 'passenger' with minimal input. We we're approaching ZZZ from the southeast and my student planned to fly around south of the field; before turning north to cross midfield at 1100 ft. and enter left downwind for Runway X. He made calls on ZZZ CTAF; when we were ten miles southeast; 5 miles southeast; 5 miles south and then overflying midfield each time stating our intentions for our pattern entry and approach. No traffic was observed in the pattern either visually or via ADS-B. No radio calls were made by other aircraft in the pattern. As we passed over midfield flying north we received a traffic alert at about our 2 o'clock less than two miles; same altitude. Initially there was confusion on our part caused by a B737 at roughly our altitude but some miles out that we thought may have been causing the alert. Then I (luckily) spotted the actual culprit: a Piper Archer roughly 50 ft. below and closing fast. I immediately took control of the aircraft as my student was unable to see the archer and initiated a climb followed by a right descending 270 degree turn; keeping the Archer in sight and returning us back on roughly a left downwind for Runway X. I am not sure how much altitude we gained or how close we came to the B shelf above but we returned to the pattern as quickly as we safely could. We then observed the Archer turn left base and perform a touch-and-go on Runway X. We extended our downwind and I made a call on the radio stating that if the Archer in the pattern was making any radio calls we could not hear them. When then flew the pattern without attempting an approach; extending our upwind leg to make sure we would stay clear of the slower; non-communicative aircraft. Finally; as we turn downwind again we heard a garbled and partial call stating that the Archer was departing the pattern.