Hi Jason,
My family is also going to Taiwan, specifically Beitou, in February. Depending on what part of February you go there is a good chance we might see one another. Our daughter is 16 months old and this will be her second trip visit her wai po and a-gong.
We love going to the public hot springs next to the Hot Springs Museum. Can’t beat the price(40 NTD I think?) and ambiance with all the local character (my mother in law goes there just about everyday). The Hot Springs Museum itself is nice to visit as well. Also, if you keep walking past the public hot springs and head up the road past the very pricey hot spring hotels you’ll find the Beitou Museum (formerly the Taiwan Folk Art Museum) which is housed in a beautiful Japanese style building and is worth a visit as well. Other nearby attractions include Yang Ming Shan for hiking and cherry blossom viewing.
Your family might also enjoying riding the Mao Kong gondola to the Taipei Zoo. We plan on going to the zoo this visit. Last year we took the gondola to Mao Kong and had wonderful meal at one of the many lovely tea houses that have great views of the city. Mao Kong is a must visit if you like tea (Ping Lin is great for tea lovers too but it’s further away). If you take the gondola it’s worth the wait for one of the “crystal cars” that have a glass bottom.
Hope you guys have fun.
Randall