I posted this a couple weeks ago and in hopes of helping others thought I would come back and follow up. (You know in case someone else finds themselves in Korea.) With the Winter Olympics coming in 2018 I would guess you will see some pick-up.
For the best sharpening I have found check out Mokdong Rink. Figure/Hockey skates are 7,000 won to sharpen. Professional level figure skates are 10,000 won. Speed skates are 20,000. The guy who runs the sharpening shop also does blade repair.
If you need new boots, blades, or boot repair and need skates above basic skills you will want to talk to the guy here -->
http://figureskate.co.kr/main/index.phpHe speaks some English and is about an hour and a half by train out of the city. He does take appointments but I can't speak for how good a job he does on a punch out. My first appointment with him personally is next week.
So if you find this post months or years from now you will have a starting point.