Anyang, huh? Well, I'm not sure, but you can try looking for a place called International Cafe Huh. It's a chain that I've noticed in Seoul and Busan. There's an open space for the general cafe and small meeting rooms as well.
TOZ is also another popular chain with great service. Very friendly staff and soundproof rooms. Fairly flexible and reasonable for booking options as well.
If all else fails, try looking around university areas. There are many smaller, independent study rooms, but these are far more likely in the city. Good luck!