Korean Search is UP
OK, you can search in Korean now (but not for how many seats are left) if you can read basic Korean. From the KOTR link, to get to the search page, go to the leftmost option on the top right, 승차권 예매, and choose the first option, 승차권 예약 -- you'll see the default is 서울-부산. Use the 찾기 to select the stations you want. (If you leave from Seoul, you need to depart 청량리 and not 서울. The latter is in East Seoul. Seoul Station has no trains to Wonju or Kanhyon!) 직통 is the default for a direct train. Obviously the other scrolls are for date and time. On the other side, you'll have to enter how many are travelling. Go to 인원정보 and use the first selector for adults (2nd for children), then hit the bright orange button and voila, the trains are listed in order, starting at the time you select (so if in doubt, choose an early time and you'll get all trains for that day from that time.)
If you are new to Korea/n and this is too difficult, you can call 1330 (and your area code) for tourist information in English.
"If you can't do something well, you might as well learn to enjoy doing it poorly."
-- from a de-motivational poster, but I find it oddly liberating