You could go to Time International Language Center for Japanese (but, it's taught using Mandarin, if you are a beginner!). Locations all over Taipei: http://www.tilc.tw/
For Chinese, the cheapest place to learn is at the Mandarin Daily News (in Taipei on Roosevelt Road): http://enroll.mdnkids.com/ChineseCourses/
You could go to NTNU (師範大學) to learn Chinese, but that's a full-time commitment, and concomitantly more expensive.
A private tutor might charge you something as low as $400 NT per hour!