PA28 Instructor reported transponder failure resulted in a NMAC by traffic overflying them.

Narrative

I was scheduled for a discovery flight in a Piper Warrior. It was a clear VFR day. The passenger that I took up is a work colleague at the FBO; so I taxied across the field from the flight academy to pick her up. While taxiing over to the FBO; everything looked to be correct including the transponder. I picked up my work colleague for her discovery flight and was cleared to taxi to runway XX. Holding short of runway XX; I did my run-up and before takeoff checklist. I verified that my transponder was still on. Soon; I was cleared for takeoff. After about five minutes I was told that I was exiting the Delta; and that my frequency change was approved. I leveled off at 4;500 feet MSL. I continued to monitor tower and do the discovery flight. I was in a left turn; looking off my left wing; when I noticed a shadow above me. I immediately rolled out of my turn; seeing the jet in front of me. I was shaken up; because I had no visual of the traffic; until after their passage. I headed back to ZZZ. Upon entering ZZZ; the tower had asked if my transponder was on. I looked down and noticed that my transponder was not on. I tried turning my transponder on four times; by clicking the power button; and the transponder was not responding. By the time I had reached the approach end of runway XX; my transponder had turned back on. I landed safely with my passenger; taking her back to the FBO; where I parked and talked with the [jet pilots]. There were two pilots who had approached me; asking if I was the instructor of the aircraft located outside. I responded by telling them yes and explained that I did not see them until after they had passed me. The two pilots claimed that we were nearly 200 feet apart from each other; and that I was at fault for the Near Mid-Air Collision. The two pilots also claimed that they saw me before over flying me; but rather than correcting their course they decided to fly over me. I believe that this situation could have been avoided; if both myself and the other pilots performed a better visual; more consistent cross-scan.From this situation; I have learned to integrate the transponder into my cross-scan more. This mental checklist will prevent further occurrences; knowing that my transponder is always on.