PA-28 Instructor Pilot reported a blockage of the pitot mast resulted in a loss of IAS; and a diversion to the nearest airport. Upon landing; the aircraft departed the runway surface and struck a snow bank.

Narrative

Inadvertent flight into unforecasted impact snow. Impact snow resulted in blockage of the pitot mast; resulting in the loss of indicated airspeed on the flight display. Upon losing IAS; I immediately turned on the pitot heat; which is labeled inoperative; and began maneuvering to the most favorable diversion airport; ZZZ. After retrieving weather and NOTAMs for ZZZ; I used pilot-controlled lighting to align the aircraft with the Runway XX center line and configured the aircraft for landing. At this point; IAS had returned and I had full operation of flight instruments. Upon touchdown; the aircraft aggressively veered to the left; uncommanded; and struck a snow bank. Upon further investigation of the aircraft; I believe the loss of directional control can be attributed to a pre-existing crack in the nose gear strut that malfunctioned on the touchdown. After the occurrence; I reflected on the situation and determined what I could have done differently to prevent or minimize the situation. While conducting the pre-flight inspection; I could have been more thorough and perhaps identified the crack of the nose gear and verified the operation of the pitot heat. Prior to departure; I should have retrieved an updated weather briefing for the return flight.