bird it might have been a single-episode drama story. If what you saw was the "Drama Special" program, that's a program that presents single-episode stories.
If you're not sure, you could always call the station and ask. When I've called recently they've been much better than in the past about being able to answer questions in English. (Apparently they get a lot of calls from English-only viewers.)
Or if you remember any of the character names, or can describe the story in a few keywords, try Googling it and see what comes up. Add "KBS" to the search.