You need a school backing you up in order to get yourself a teacher's license. If the school can't do the paperwork then your own chances of doing it will be slim to non-existent.
Many schools do not actually know how to get licenses and work permits for foreign teachers, or do not have a member of staff who has ever done it. In this case things can get very drawn out with the application being postponed indefinitely. If you're the first or only foreigner in a school, good luck.
The actual process need not take a long time. The important thing is to get the teacher's license because that will enable you to file your work permit application, which is then enough to extend your visa.
The process shouldn't take more than a month according to several school admin people that I have spokn to.