"Dal Ja-sshi" actually would be a perfectly fine way for her colleagues or her boss to call her. The "--sshi" designation elevates the first-name-calling into a more formal territory, like "san" desgination in Japanese. It would be just as polite as calling her "Miss Kim" or something, although the fact that they know her name does put them on a more familiar territory. Her close friends, on the other hand, would call he just "Dal Ja" or "Dal Ja-ya" (if they're calling out to her.)

Hope that helped! (It cracks me up so much you're learning Korean through kdrama! Reminds me of the time when I was learning English by watching a gazillion rerun of "Family Ties" 20 years ago.)