Reference letters, although not difficult, can take some time.  If you request a reference letter from someone you will be dependent upon their ability (or willingness) to take time out of their day to write it, print it, sign it,  scan it, and email it to you.  If you've asked a former boss to provide a reference letter and they truly are busy (not all bosses are created equal) then it may be very difficult for them to take time out to do it.
I figured this would be the case seeing that my two references came from a CIO of a medical devices company, and a Managing Partner of a management consulting company.  What I decided to do was to write the reference letters for them.  I spelled out what my functions were when I worked for them, my title, some specific responsibilities that may be teacher-friendly, and some personal statement about why they would feel I'd make a great teacher.  After drafting the letter I emailed it to them.  This way, all they had to do was have their secretary print out the letter and have them sign it.  After that, the secretary could scan it and email it back to me.  Piece of cake.
One of the letters I wrote was a bit short according to my recruiter so I reached out to a director of a homeless ministry I once volunteered at.  I did the same thing, scripting the letter and just sending it to them to have it signed.  In all cases I had the letters back within 1-3 days.  Done.
I would recommend making reference letters at least one full page.  No extreme blank gaps or useless fillers like images or logos.  If you do this, you'll have great reference letters in no time and be teaching in Korea quicker than you can say "kimchi".