One method that often works with unusual encodings is copying the text to the first page of a new
Sigil project, and saving it as an ePub. (Make sure to select the correct language in Sigil (Tools > Meta Editor... > Language) before saving it.
(If your text files are very large, convert them to .html files and add them directly to Sigil.)
If the Korean characters look OK in Sigil, you can convert the ePub file to a mobi file with either Calibre or
Kindlegen, which accepts ePub files and tends to be a bit faster than Calbre.