I'm not the one who knows about
a formal exchange student aplication letter to korean university
. However, I have a few to share.
First Para: talking about your interest in student's exchange programme, which might be relevant for your future career.
Second Para: background education. Here you can say one-two personal stories you face when facing this study
Third Para: Responsibilities related to your study, or extracurricular activities if any.
Fourth Para: Closing statement indicating your expecting for the reply.
Hopefully, it works :)