Pilot instructor; in a DA20; reported a loss of nose wheel steering and brakes causing a taxiway excursion.

Narrative

We had stopped in the solo runup area to grab the radio bag and I had told the SP (Student Pilot) that I would just take care of getting us back to the ramp. I started the engine and got clearance to taxi back to the ramp via 1 runway XX hold short of YR. As I was taxing us down Taxiway 1; at approximately 15 kts; the airplane started to go to the left of the taxi centerline; and I pushed the right rudder and got no response and then I put pressure on the right brake and didn't get any response from it. At this point I realized that we were going to go off the taxiway. I got up onto both brakes as hard as I could to try to see if I could get the airplane stopped before it went off the prepared surface. I realized I was going to be unable to stop the plane in time and pulled back on the control stick full aft and pulled throttle to idle. I was unable to get mixture on first attempt but did end up getting the mixture to idle cutoff. SP got the ignition switch to the off position for me. I made a call to ground control of our situation and then shut off the Gen/Bat master switch. Aircraft went off taxiway approximately 15 ft to the south of taxiway 1.