If you're mountain lovers, I would strongly recommend you to plan a trip to go the Eastern Taiwan, or even a trip around the island.
A cab with guide driver cost around 260 USD per day. If you have many travelers, including young kids, this would be convenient and economic.
1.(新北投Beitou): This is a famous tourist attraction for hot springs. Many hotels you can choose depending on you budget. There is a 北投圖書館 well-known because of its green and beautiful architecture.