Flight Instructor reported the brakes failed during taxi resulting in a taxiway excursion and the propeller hitting a taxiway light.

Narrative

I was acting as Pilot in Command of a discovery flight with a student at ZZZ. The airplane we were flying; a Cessna 172S; Aircraft X; was preflighted and ready to go. After startup; my student and I tested our brakes. My side felt spongy but still worked. During the engine run up; I had my student help me hold the brakes to keep from rolling. After the run up; we were instructed to taxi to Runway XXL via Taxiway XX for departure. The airplane taxied normally; and we departed without incident.Upon returning to the airport; wind conditions increased significantly. We departed with winds 150 at 10 to return with winds 160 at 13G26. I elected to use 10 degrees of flaps and approach at 75-80 kts. instead of the typical 30 degrees of flaps and 65 kts. airspeed. The airplane was moving too quickly to safely stop and make Taxiway XY; so I rolled down to the end of the runway. The airplane made the right turn safely and we taxied back to parking via Taxiway XX. Taxiway XY was the first entrance to the ramp; and that was the entrance I was planning on using.At this moment; the left brake pedal on my side (right side of the airplane) had no pressure on the pedal and was doing nothing to slow the airplane down. The airplane turns with differential brake pressure; to which I had none to make this turn. I asked my student to get on the brakes on the left side to help me; but that was too little too late. We departed the pavement and stopped in the grass short of a taxiway sign; but not before clipping the blue taxiway marker light with the propeller. Miraculously; only the propeller was damaged; and only slightly. In hindsight; I could have taxied further along XX to the next entrance to the ramp and had my student help me; but in the moment I had no idea that my brake would completely fail. Everyone walked away safely; no one was injured. Shaken; but otherwise okay.