I love South Korea and was lucky enough to be stationed there twice. If you can arrange it, a must-do is a tour of the DMZ. It was easy to arrange for military members because the bases all do tours and take you to some special places like the Panmunjom Joint Security Area where the negotiations between north and south take place. You can even go into the negotiation building that spans the demarkation line and walk "into North Korea" by walking to the far side of the building, It's very eerie to have North Korean soldiers around watching you and South Korean soldiers guarding you while there. Some tours also take you to go inside tunnels that were discovered from the North burrowing under the DMZ into South Korea and some great viewing areas to look down into the DMZ no-man's land and propaganda villages that the North has created along the border.
Joint Security Area Panmunjom
Go hiking in a South Korean park. Hiking is an enjoyed activity and the parks are all beautiful. I went hiking many times in the mountains around Seoul and it was really sobering to be walking on a trail and come across a manned anti-aircraft gun pointing North. You won't see that in other places but in those mountains north of Seoul.
Go see the Korean War Museum in Seoul.
If you can spare the day or two from your itinerary, go see Jeju Island at the southern tip of the Penninsula.
And if you can add several days to your trip, while you are traveling so far already, why not see some very special places only short flights away from Seoul like Cambodia's Angkor Wat which is an amazing place as special as the Pyramids of Egypt. Also Thailand or Vietnam are great places to spend a couple of days.