In this scenario it might have been best to get a D-4 visa since you are learning Korean at Seoul. I'm not too sure if it is a D-4 so you should ask your local Korean embassy for clarification.
(Full link) http://www.studyinkorea.go.kr/en/sub/overseas_info/guide/guide_visa.do
Visa for General Training (D-4): For persons who study Korean at a university-run language school, persons who study in a qualified educational institute or an educational institute other than an academic research institute, persons who get technical training at national or public research institutes, training institutes, bodies etc. persons receiving payment from research institutes or who have industrial training system qualifications
Visa application form or Certificate for Recognition of Visa Issuance
Processing Fee: Amount corresponding to USD 30 (Single-entry) or USD 50 (Multiple-entry)
-Verification of School Entrance or Enrollment (Verification of Training)
-Financial Verification (more than USD 3,000 in domestic bank account or certificate of currency exchange) or documentation under a university academic exchange agreement
Identification Verification (for those who cannot verify their financial ability for tuition and living expenses or upon the request of Justice Ministry)
Certificate for Confirmation of Visa Issuance: In order to shorten the issuing period and simplify visa issuance procedures, an immigration office head can issue a certificate for recognition of visa issuance after examining the applicant's documents upon the request of a sponsor in Korea. Then, the Korean missions overseas will issue a visa based on the certificate. The certificate is required except for special cases. The applicant or sponsor in Korea can apply for the certificate at the nearest immigration office (or its branch) with required documents. When the certificate is issued and sent to the applicant, then the applicant can get a visa by submitting the certificate to the nearest Korean missions overseas. The certificate is effective for three months and valid for a single visa application.
Just remember that you will have to travel outside of Seoul to apply for the visa.