This is an option I had never considered! The Taipei Arena (the beautiful arena at the corner of Tunhwa North Road and Nanking East Road, Taipei) boasts an iceskating rink, open to the public, and with rentals available. At $190 for two hours, plus $80 for rental of equipment, it is a good option for these rainy, cold days. I am not sure exactly what the lower age limit is, as the site only mentions that children under 7 need to be accompanied by an adult.
The English site is pretty comprehensive. Just run your mouse over the informational headings to find out what you need to know.