A Tower Local Controller reported they cleared two aircraft to land and one to takeoff before realizing a previous aircraft exiting the runway was stuck in the snow and not past the hold short line. The reporter stated ASDE X indicated the aircraft had cleared the runway.

Narrative

I was working Local Control West departing and arriving aircraft. The weather was IFR with a 1/2-mile visibility; snow; 3000-4000 RVR's and Braking action of good on Runway XXL. I had the IFR turn releases and call for release on the straight out departures. I had just cleared Aircraft X to LUAW (line up and wait) on Runway XXL and told him about traffic on about a 6-mile final waiting for an aircraft that departed off of the east Runway to go west bound. I cleared Aircraft X on a 280 heading for departure. Right after Aircraft X read back his takeoff clearance Aircraft Y who was on a 4-mile final said they were going around. I initiated go around instructions to Aircraft Y. After that I went straight to Aircraft X and canceled his takeoff clearance and to hold in position for the aircraft that was going around. After taking care of Aircraft Y and shipping him to departure; I went back to Aircraft X and asked him if he needed more time on the Runway or to exit to rerun some numbers. He said he needed to exit and make some phone calls. I exited Aircraft X to turn right at [Taxiway] X and he read it back. Aircraft Z was next in line so I told Aircraft Z to LUAW. Right after that someone said that the aircraft that just exited need to move up a little bit more. Not sure who said it. Went back to Aircraft X and said can you move up a little bit more and he replied yes. I then went back and told Aircraft X to taxi up XY and hold short of XZ. Aircraft X read it back and started to move. I verified all of the movements on our ASDE radar. The Aircraft X showed he was clear of the Runway at all times when I lined up Aircraft Z. Even after Aircraft X read back the taxi instructions the aircraft showed it was moving on our ASDE. After showing movement on the ASDE I cleared Aircraft Z for takeoff without incident. I believe after Aircraft Z departed Aircraft X said he was stuck and could not move. I believe that [Taxiway] X was never cleared of snow prior to this event which led to Aircraft X to get stuck in snow. I landed two aircraft Aircraft A and Aircraft B with the information I had showing Aircraft X was clear of the Runway. I asked for on Operations vehicle to go and check on Aircraft X. It was after Aircraft B landed that Operations said that Aircraft X still had part of his aircraft (tail portion) still 20 to 30 ft. past the hold short line. After that I coordinated with the tower supervisor and Operations that I was closing Runway XXL until Aircraft X was clear of the Runway. I recommend during snow and low visibility where you cannot see the Runway or exits that if we have to cancel or clear an aircraft off of the Runway that we take them down to an area like [Taxiway] Y and beyond where the high-speed exits have been cleared of snow removal. Also; to get a call sign of the person giving me information about the aircraft who was stuck. I should have errored on the safe side and cleared the aircraft from [Taxiway] X by taxiing him north or south to make sure he was clear of the Runway. I should have taken the Aircraft X into account knowing that this is a longer airframe than the other Airbuses.