Single engine instructor pilot reported an NMAC while in the traffic pattern at a non-towered airport. The instructor and trainee were on left downwind when an approaching helicopter from the right flew through the final approach course and evasive action was taken. The instructor was unable to contact the helicopter pilot.

Narrative

I was performing landings at ZZZ with a student in Aircraft X. On the left downwind for Runway XX; I noticed an aircraft at the same altitude approaching from the north at 4 miles. I assumed they might be entering on a left base for the runway; but they hadn't made any radio calls. As the student was approaching left base turn; the aircraft was approximately 2 miles; and roughly same altitude. I saw it was a helicopter and assumed it was a Company helo based on the north ramp which may be turning inside of us to land on the taxiway.The student turned base; and the helicopter continued on a southerly heading inside our pattern and approximately 200 ft. below us. As the student started to turn final; I could see on ADS-B that it had not changed course; and as I reestablished visual; I could see it was not a Company helicopter so I initiated a go-around and climbed above the aircraft and continued on runway heading for another lap while the helicopter continued on course without correction. Aircraft appeared to be Aircraft Y indicated on ADS-B. I tried to initiate radio contact; and received no response three times. The incident was caused by the helicopter transitioning southbound at and below traffic pattern altitude within 1-2 miles of the arrival end of the active runway while additionally not making radio calls or being on the CTAF frequency.