PA-44 Instructor reported a near miss with a helicopter while on a go-around due to landing aircraft not clearing the runway in a timely manner at a non-towered airport. The Instructor performed a maneuver to fly under the helicopter; over the airport surface; then returned for a safe landing.

Narrative

We had an uneventful flight up to our arrival to ZZZ. Tower was closed and we were following the noise abatement traffic entry procedures. We followed traffic inbound to ZZZ to overfly the airport at 2;900 MSL for left teardrop into the right downwind for Runway XXL. Traffic we were following was a Cherokee. We turned base to final behind them and at the minimums callout the aircraft was still on the runway. After a brief moment I called a go around to my student. My student initiated the go-around procedure; in the incorrect order which I was coaching her through since this was her first go around in the aircraft. We had a positive rate of climb; the gear was up; and the frequency was too congested to make a radio call reporting the go around. Out of the corner of my eye; outside the left front window I caught visual contact of a large helicopter Aircraft Y which appeared to be at a slightly higher altitude; traveling northbound (seemed to be following Taxiway 1). Without time to react I identified this as an impending collision and immediately pushed forward on the yoke lowering our pitch well below the horizon to pass under the helicopter. The helicopter passed overhead. After which I re-established a climb; extended upwind and made right traffic landing back on XXL a few minutes later; uneventfully.This safety report is to report on this close call; as well as to highlight a deeper issue which occurs here at ZZZ. When the Tower is closed; during a high traffic time of the day operations in and out of this airport become very unpredictable. Non-local traffic often has no idea on the tower closed procedures and comms are usually heavily congested. Everything is busy and while dealing with a student it is very difficult to safely navigate at times. During this incident; comms were congested with several traffic conflicts and a non-local plane which had no idea how to enter the pattern which made it very difficult to keep tabs on every aircraft at every moment. On top of that; helicopters fly unknown (to fixed wing pilots) arrivals and departures; which take them over the airport right at go-around altitudes.ZZZ airspace is dangerous when there is no ATC operating. Tower closed at XA:00 time on this day; presumably due to staffing issues.