I essentially had this same problem at my school, except for that she wasn't one of my students (she was a third grade student, and I don't teach the third graders).
We have a councilor in our school, and her job is to help people like this. The councilor was/is part of our department so she would be in our office all the time. All of a sudden she always came to our office (during class times) with another student.
The student seemed way more timid than any of the other Korean people I've seen... and I was always wondering why she was in our office instead of being in class.
Eventually I asked about it and found out that he parents separated and that her father was/is abusing her. Because of that, just like the OP's student, she was afraid to talk with men, or be in the same room 1 on 1 with a man (I can understand that one more than the speaking one).
To the original poster:
Luckily, you have a very strong asset that nobody else in the school (probably) has.
You are foreign. Because of this, the girl might not look at you as a typical male... even better, she might actually want to get to know you.
This was actually the case with the girl at my school. She hated Korean men so much that she said she would only marry a foreign man... but she can't speak any English...
She said this to my co-worker at the time, but I was sitting beside her and I understood what she said. I told her that I could teach her English one-on-one (in an office with other people)... and her face literally lit up. The councilor and other teachers in the room were so shocked because she started speaking to me instantly (in Korean, but who cares, thats not the point of all this). For a month or so she would come and see me a few times during the week and I would teach her one on one. It even got to the point that sometimes we were the only people in the office and she didn't seem to care at all.
It heavily depends on your situation... but if you can speak some Korean (or she can speak enough English to understand you if you were to teach her one-on-one)... you could try to talk to her about something like that.
Realize that you're foreign, which might make you look completely from other Korean men to her.
