Flight Instructor reported the right engine failed while exiting the runway after landing resulting in the student losing directional control of the aircraft several times.

Narrative

After a long float to a long landing we exited at the reverse high speed taxiway; we were given clearance 'Taxi to parking on [taxiway] 1 cross RWY X to [taxiway] 2'. Shortly after starting on [taxiway] 1 the right engine failed causing us to veer to the right. I immediately applied left rudder but this aircraft has no brakes on the co-pilot side. As the PF began to realize there was an issue; the RPM was still too high ~1250 to control with rudder alone; the PF applied brakes but that wasn't keeping us near the TWY centerline; alas I was along for the ride. We were stopped in the middle of 1 on RWY X because the PF was not controlling the plane single engine. I suggested we start the right engine which started pretty easily and now we were turning to the left because the PF had 2000 rpm on the right engine; we were approaching the edge of the taxiway and the taxiway lights between [taxiways] 3 and 4. The PF then asked if we were cleared to 4 which I said go ahead because we need to get to the ramp instead of ping ponging on the taxiway. Ground called and asked if we were on the radio; we acknowledged; I advised ground that we exited due to engine failure and controllability issues; the ground controller told us he would let us slide this time but almost had us call the tower.