Once the kids are at toddler stage, you can add a few museums and eateries to the list, too.
The kids land area in the Museum of World Religions (Yunghe) has a wonderful magical playland with a small library, and kids’ bathroom facility. They have a one-year membership that is $1500 and gives you access to the museum for a year, and you can take vistors to Taipei with you for free, too.
As Isha mentioned, the Children’s Science Museum up in Tienmu has a good play area for kids in the basement. Kids can also get excercise going round the atrium, and there are places to eat inside. They have a one-year membership available, too.
The Children’s Art Museum in Tienmu is a wonderful little adventure for children from toddler age through to about 10 years.
Taipei Storyhouse in Xinyi area is free and has three floors of interactive displays that children can enjoy.
I know some parents also like Lollypops for very little ones. (Family restaurant up in Tienmu.)
I really like the MacDonald’s playarea in their Reifang branch. If you are out in the North-East Coast area and get caught by weather, the 2nd floor of their MacDonald’s is usually empty and the play area is quite big. They have a big car park out the back.
It’s also easy to forget things like a short rainy-day train ride to break up the monotony of the weather. Even just from Taipei to Keelung or Panchiao and back, with maybe a stop for a drink or treat, can break up a day. Dad used to take us on rainy-day drives to buy an ice-cream, and they are wonderful memories for me now.