A friend of mine has recently opened a quaint cafe in Wonju, South Korea (1 hour from Seoul) called Apple Farm / 사과농장. It serves hot/cold drinks and light food daily in a cozy relaxed atmosphere where even well-behaved pets are allowed:
All the interior design was done by the very talented proprietor, Seon Mi Park (박선미).
While Wonju might not be a big city its proximity to Seoul makes it popular with commuters and students from nearby universities.
Do check it out if you happen to be in the vicinity! I’ve heard great things about the home-made chocolate cake
강원도 원주시 중앙동181-3 번지 2층
2F, 181-3, Joong-ang-dong, Wonju, Gang-won-do, South Korea
Open daily 11am ~ 10pm (closes 6.30pm Sat)