I have a letter of invitation to teach at Kunming International Academy, Yunnan Province

You can apply for work Z type for that. You should first prepare those documents including your passport and copy of its data page, a completed application form, a recently-taken passport-style photo, and a work permit from your employer. Then, submit the application to the Chinese embassy or consulate in person. You can find someone else to do that on your behalf if you cannot go there in person.