December in Taiwan is okay as there are still so many things you can see and do, just have to deal with the weather. But with kids in tow, it can be quite a hassle. Best to stick to the summer months if you intend to take your kids on a holiday to Taiwan. You can consider other destinations like Thailand or Vietnam perhaps?
The weather is so unpredictable these days. I went on March this year, out of 7 days, 4 days were downpour at about 10degree to 17degree at night. While my extended family members went on December, the weather was less chilly and rainy comparing to March. :)
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Yes, December is quite damp. I remembered showers during evening almost alternate days. June is still the best month to visit Taiwan.