I am sure there are things to do nearby the hotel, but I haven't seen it yet.

How about taking the subway Blue line up towards Jongno 3-ga station? Take exit #1, walk straight out (West direction). Then pass Tapgol Park on your right side, and then head up Insadong street? Insadong is a street that is diagonal to Tapgol park and is a great place to see various arts and craft items, plus souvenir shop. Always a fun place to spend some time.

If you have time, and don't mind eating vegetarian food, there is a great temple food restaurant called Sanchon. I would recommend the dinner there as the price includes a free performance of korean dance and drumming. Maybe you can ask the concierge at the hotel to make a reservation and better yet, draw a map on how to get there. It is located in Insadong neighborhood but it is way hidden off the main path. All I can say is if you see the curly friend potato stand, it is a few turns behind it.