When I lived in Muzha, I often lamented the lack of good indoor swim facilities for children. Now, with the builiding of the Wen-Shan Sports Center, there is a very nice child-friendly pool in the area. Like all government-built sports center, it is well-priced, too.
You can park in the basement of the sports center, or enter from the first floor if you walk or take the bus. You buy your tickets for the pool on the first floor. If you have forgotten your goggles or swimcap, you can pick something up at the little store. Head to the 3rd floor and put your card into the slot to enter the pool. Right there on your left, there are two family changing rooms. When we went last week, they were very clean and in good order. Take your gear to the pool side and put it into the pay lockers ($10 per locker). This pool has the system set up in a very family-friendly way (no running in and out of changing rooms like older sports centers).
The main pool has a large graduated slope into the pool, which is ideal for children learning to swim. Just make sure the kids don't climb up onto the rails, or the lifeguards will come to get them! There are two kids' pools - one 60cm deep and one 80cm deep. In those pools, children can practice swimming. Unlike pools I'm used to at home, there is no opportunity for playing or tomfoolery; it's all very serious. The signs say, "This pool is for practicing swimming. Do not play around." So, it was a bit of a serious swim. A fun, serious swim. My kids had a great time, and are begging to go back.
*There is no soap or shampoo in the showers. It's best to take your own, as they currently only sell special shampoo at TWD450/bottle.
On the first floor of the facility, there is a nice little ball room and play area, and a coffee shop. There are other facilities and classes, but with the weather as it is, I think the swimming pool is the place to be!
Address: 台北市興隆路3段222號 Shinglong Road Sec 3, No. 222 (Just south of Wanfang Hospital)
Chinese Website: http://wssc.cyc.org.tw/
Here is a list (Chinese) of classes running at the moment. They have swim classes for children aged 3 and up.