Seconding the recommendation for Healer, it's one of my all-time favorite Kdramas. Excellent story, excellent acting, just excellent all around.
While You Were Sleeping has a few mystery elements to it. It's a fantasy legal drama where the main characters have prophetic dreams, usually relating to current cases, and they have to figure out the truth of what really happens. Argon is another one you may want to consider, about investigative journalists. Oh My Ghostess has a pretty good mystery going throughout the series - the titular ghost character doesn't remember how she died, and the m