On our couple final days, we leisurely spent most our time in the city.
We managed to see the royal guard changing ceremony at Gyeongbokgung Palace in the morning. We also visited the palace museum which surprisingly had quiet extensive collections.
We also went to Bukchon Hanok Village, the Korean War Museum and strolled around Itaewon.
I personally enjoyed the Korean War Museum. It was huge with extensive history not only about the Korean War itself but also Korean History in general as well. I suggest to spend at least 3 hours there.