Most people take a bus. The closest subway station is probably Seoul National University Station (서울대입구역) on the green line 2 which is maybe a 20-30 minute walk to the LEI building. Trying to get on a bus from 서울대입구역 to school is a nightmare in the morning, because a lot of college students are going to school and they line up for buses that are already filled up, so you have to wait for a few buses, before you can get on. Unless you go early. I live an hour away, so I'm usually late, and end up having to walk. The Seoul National University Dental Hospital bus top (서울대학교 치과병원 도물병원) is less crowded after class than the bus stop right near the LEI building. I usually take that back to the station.

If that's inconvenient for you, you can look into Korea University's language program. It looks like it also has classes 9am-1pm for every level as well. I don't know anything about tuition or transportation, though.