Air carrier Captain reported a loss of nosewheel control when entering the ramp due to icy surface conditions while taxiing to park at night at a non-towered airport. The Captain could not maintain directional control and received a tow safely to the gate with a tug.

Narrative

Lost all nosewheel control authority on GFK ramp and had to be towed to the gate. The runway and taxiway were clear and essentially free of contaminants; so I elected to shutdown the left engine; not realizing (or able to see from that point in the taxi) that the ramp was still mostly iced over. Upon entering the ramp; the nosewheel would not respond to directional inputs multiple times. The aircraft also skidded uncontrolled multiple times; the ramp conditions were very dangerous for taxiing. We decided to restart the left engine to provide more directional control; however it inexplicably failed to restart on two attempts (info only writeup completed and Maintenance Control notified). Eventually it became clear that we could not move under our own power; safely or otherwise; so the ground crew attached a tug and towed us to the gate. The left engine restarted on the third attempt; but this was already after the aircraft was attached to the tug.Of note; due to the irregular circumstances of getting attached to the tow outside of the normal process; and likely influenced by the lateness of the evening (the flight suffered from multiple compounding delays); I almost forgot to secure the nosewheel steering! This was caught seconds before the aircraft was towed. The entire process to get us to the gate took approximately 25 minutes. Cause: Premature shutdown of the left engine (decision was based on conditions experienced at the runway and taxiway; and therefore expected to be on the ramp); unsatisfactory/unsafe ramp conditions (ground crew noted that the airport operations authority should have sanded the ramp under the circumstances; which did not occur; but this was completed by the time we operated outbound the next day); inability to restart the left engine.Suggestions: I cannot control ramp conditions; therefore in conditions where there is snow/ice on the airfield; I realize that I should keep both engines running until ramp conditions can be visually verified. While airport authorities paid sufficient attention to the runway and taxiway; the ramp was completely neglected; and it should not have been.