B737 Captain reported a temporary loss of control while taxiing due to nose wheel steering failure. The aircraft was stopped on the ramp and towed to the gate.

Narrative

We taxied out from ZZZ1 with no issues on dry taxiways. We landed RWY XYR ZZZ which was a wet runway and turned off high speed Taxiway 1 (left turn) at 35kts no issue. Taxied south on Taxiway 2 as we entered the right turn elbow (30 degrees) of this taxiway at 15kts while applying light braking the nose gear steering (tiller) had zero response to the input and the nose gear began a violent shutter. I attempted to use up to full right tiller with zero response and had to apply heavy braking as we are already starting to leave then centerline. The aircraft did respond slightly to the hard over right tiller once it had almost come to a complete stop at maybe 2kts and started a slight right turn. We ended up facing about 20 degrees left of parallel with the intended direction and then left main gear was about 8 feet from the east (left) edge of the taxiway. At this point we notified tower we had an issue and we needed to stop there to asses the situation. He said that with his binoculars it looked as if we had a flat left main tire. We then called for ARF and airport authority to come to the aircraft followed by me calling our dispatcher whom put us into a three way call with maintenance. The ARF arrived and said there were no abnormal heat signatures with the brakes and that the tires appeared to be inflated properly. At this point we waited for maintenance to arrive at the aircraft to be sure there was no damage and that the aircraft was safe to continue taxi to the gate. Tire pressures were verified normal and the steering tiller was verified to work with a full right and left from the parked position. At this point maintenance and I agreed the aircraft was safe to taxi so we took minimal speed taxi with the slight right turn onto the centerline without issue and headed south on Taxiway 2. We made a 45 degree left turn at 15kts; followed by a 90 degree left turn at 8kts with no issue. Once we began the first right turn 90 degrees at 7kts the nose gear had zero response to tiller input and started the same violent shutter so we came to a complete stop. From the parked position I began a taxi as slow as possible to complete the right turn without issue. The plan was to cross RWY XYL and get to a spot that would keep us safe from landing and taxi traffic where we could call to have the aircraft towed. We made the left turn onto Taxiway 3 with no issue and ground parked us at [the gate]; the right turn into [the gate] was done at around 3kts. From this point we contacted maintenance for a tow to the gate. The entire airport surface was wet with very light rain and in my 15k plus hours I have never had an issue such as this to include taxiing on ice; snow and non grooved airport surfaces under moderate to heavy rain. It is my recommendation that before return to service this aircraft be tested under similar conditions to ensure the safe operations going forward. Steering failed as the turn was initiated and required an immediate stop.