OCAC organizes a lot of youth camps that include culture and Mandarin classes. (aimed at younger age groups, you can look into that)
I also had attended the Tzu Chi University Summer Mandarin Language Camp when I was 18, the age group is from about 13 to 24 and it included culture classes (Chinese calligraphy and painting, flower arranging, Chinese music), sports classes and Mandarin classes based on the level. I think you have to know someone in the Tzu Chi foundation to send a child though, because they don't accept just anyone.
The Mandarin Daily News also has a language training center that aims at children. http://study.mdnkids.com/news_e.asp
As a 22-year-old who just moved back to Taiwan, I can speak and understand Chinese, but my reading and writing are in a low level, so I will be attending National Taiwan Normal University Mandarin Language Training Center (國立台灣師範大學國語教學中心) http://22.214.171.124/mtcweb/index.php?lang=en
which accepts different age groups, but I think they focus on teenagers to college graduates, not sure about kids.