Air carrier flight crew reported a loss of control during taxi after landing due to ice contamination on the taxiway. The Captain stopped the aircraft and the airport sanded the taxiway; no damage occurred.

Narrative

Coming into to ZZZ on Runway XX. I chose to utilize flaps 45. Little after the final approach fix; I disengaged the Autopilot. The winds were very strong gusting. We were being rocked by the winds. Pretty strong turbulence on final. We hear over the radio that Aircraft Y got stuck after landing. Which alarmed me cause they literally just landed; that I thought they were stuck on the runway. We asked ATC if we were still cleared to land. They didn't respond. I tell my FO (First Officer) to ask ask again. I am about to call a go around as I didn't know if the runway was clear. I look out the window to make sure the runway is clear. ATC finally advises us the runway is clear. During this task saturation of fighting the winds; trying to stay level; make sure Aircraft Y is off the runway we got a huge spike 15 plus kt. increase on the speed followed up by a quick 15 plus kt. decrease. After landing we start taxiing to the gate. We then too; have the same issue as Aircraft Y. We begin sliding on the ice. I chose to stop taxing and told the ground operations they need to either tug us into the gate or lay sand out; which they chose to lay sand out. Which took another 40 mins. During this time my FO advised me; they believe we oversped the flaps by 2 kts. We advised Maintenance and they said they would get it checked out. Maintenance later advised me they were able to clear this out. Suggestions: We did brief the plate well. I gotta keep the scan of the instrument going continuously; doing that better.