C172 flight instructor reported ATC advised their transponder was not showing during approach. During troubleshooting they lost radio communications; continued the approach without radio communications; resulting in a NMAC.

Narrative

On approach eight miles north of ZZZ airport; we were in contact with ZZZ Tower. Tower advised us that our transponder was not showing up. We were advised to recycle the transponder. I was not as familiar with the transponder; so my first thought process was to recycle the avionics switches. As I recycled the avionics switches we did not get comms back or our transponder. I immediately took flight controls away from the student; and tried troubleshooting by recycling the avionics switch again; which again made nothing work on the avionics busses. My instinct was to divert us to ZZZ1 because we had no communications; and we should not have gone to a busy Delta airspace environment to look for light gun signals.As we proceeded to divert; I was scouting out the area to see if there's any traffic in the pattern for ZZZ1. I saw a light sport aircraft enter a downwind for Runway XX. I decided to follow the light sport aircraft to the runway. I was able to see that my COM1 was lit up; but I could not hear myself talk over the comms; or hear anybody else transmitting on the comms. I tried transmitting in the blind; but I could hear nothing. I could not even hear my student or my voice over the radio's during the whole entire situation.Eventually; we turned base to final; where; as we turned; a C172 was above us less than 300 feet away that had established themselves on final. My instincts were to break right immediately away from the traffic above us on final. As we did a right 360; the C172 we deviated for had touchdown; and we proceeded to touchdown after them and taxi to the general aviation ramp at the airport.At the airport; we contacted Company's Dispatch and Manager over the phone to let them know that we were safe on the ground and we had a lost comms [problem]. Afterwards; I had contacted ZZZ Tower on the phone to advise them that we had touched down safely at ZZZ1 and we were having issues with our comms and our transponder. While on the ground and after my phone conversations; I later turned on the master switch and the avionics while on the ground to see if we had the same issue. Our comms came back online.45 minutes after we had touchdown; we contacted ZZZ Tower over the phone to let them know that we were planning on coming back and that we had our communications working again; but we did not know if our transponder was working. They told us that we could come in and that they would work with us to get back to our home airport.We flew back to ZZZ; and Tower advised us that our transponder still was not working. We made a safe landing; and I squawked and grounded the aircraft to our Maintenance Department for having an inoperative transponder.