One of my English friends took classes at Chinese Culture University. He said you can go into the office and ask about 1-on-1 tutoring.

I did 1-on-1 classes for one year at Chinese Education Center, near MRT Chiang Kai-Shek Memorial Hall. My teacher was pretty good. We'd do 30-minutes of open conversation completely in Mandarin Chinese, then a structured lesson. She had supplementary handouts that were helpful. One I remember was how to order at Taiwanese breakfast shops where they had danbing omelettes and those awful sandwiches on white bread with the crusts cut off.

PM me if you have questions. I lived in Taipei for 3.5 years.