You can easily head into the city by train or limousine bus. To avoid traffic, I'd recommend the train. With a two year old, it might be better to take an express train as you have a guaranteed seat. The all stop train is more frequent and only 10 minutes longer, but it is subway style so a seat isn't guaranteed.

You can take a taxi from Seoul Station to Gwanghwamun. There you can explore Gyeongbokgung Palace, Insadong, Hanok Village etc. See the main sights, try some Korean food, shop for anything. This is also the downtown area so it is a good city vibe as well.

You don't need to get a hotel for a shower. The airport has free shower rooms airside to use like you'd find in an airline lounge. They're on the 4th Floor Rest & Relax area. There's also a traditional Korean sauna in the basement landside.

BUT if you want to go to a sauna you should try Siloam conveniently located behind Seoul Station. They have excellent facilities, resting areas if you want a nap and even very good food! Around 10$ per person for entry.