I also recommend not living in Taipei.
I would recommend Taitung City or maybe Dulan near Taitung instead. Your money will go a lot further and the schools will be more relaxed than hyper-competitive Taipei. Kaohsiung City and Yilan are other possibilities. In Yilan, you might consider living in Toucheng so that your kids can go to the Humanities Primary School there. blog.ilc.edu.tw/blog/blog/27589/ … 253/469943. Toucheng/Yilan are close enough to Taipei so that you could get into the big city on the weekends but still enjoy a laidback rural/lifestyle.
If you must live in Taipei, I would recommend one of the “mountain schools”:
Datun: w2.dtps.tp.edu.tw/english/introduction.htm
Hutian: htes.tp.edu.tw/
I also suggest that you read the earlier posts on this blog: talesfromthebeautifulisle.blogspot.tw
The author (who posts here) describes her experience in having her son go to a Taiwanese elementary school. I think he was about eight. It sounds like it was a bit rocky at first but that it worked out.
Good luck!