I always go to Coheart in Susong-gu. The owner (I don't remember her name) studied in the UK, so she speaks English quite well and knows how to deal with foreign hair. She has done hair for 2 of my friends in the past and they have always really loved the result. I usually go to her second in command, Minjae. While he doesn't SPEAK English well, he understands it and is extra careful to triple check with clients that he is doing exactly what they want.
They are a little pricey for a Korean salon 25,000~30,000 for a shampoo, scalp massage and style, but it is apparently a more luxury place, according to my Korean friends.