The first thing that came to mind is
Romantic Doctor Teacher Kim, which at one point played Material Girl by Madonna and Time After Time by Cindy Lauper. And
A Wife's Credentials has Turn! Turn! Turn! by the Byrds. All of which you might not be familiar with if you're too young! But I'm old so I'm not knowledgeable in contemporary music, at all :D
(Oh, and since @zeamays has pointed out various "interesting" English songs in Kdramas, I can't help mentioning that the most surprising song that I've heard in a Korean drama is You Showed Me by The Lightning Seeds, in Evasive Inquiry Agency. Which is a British band and not really famous, I think.)