new visa

Hi!
This summer I'll be doing a Research Project for a company in Xiamen, but after my project I would like to visite Korea before I'll visite my family and relatives in China. I heard from my company that a working F is only possible for a single entry. So told me to apply for a new one while I'm in Korea, it would be for a L. Is that possible? How long will that take and do I need an invitation letter from my relatives? I self have a Dutch Passport.

Thank you