There is / was a building with one of the bath house / sauna signs at the back of Seoul station. Never used it so no idea what it's like.
If you haven't already booked a train to Seoul you might want to consider getting the bus to Incheon from Nopo as the seats on the buses are the huge, super comfy, lazy boy style with a huge recline and get your sleep there. Also if you book early online you can get a single seat all to yourself. Once at Incheon catch the connecting bus (20 mins) to Gimpo. (better check times of connecting buses depending on what time your flight is.