For two days, may I suggest some combination of the following:
(1) Chiang Kai-Shek Memorial, 228 memorial, Presidential Palace (walk by only–can’t go inside). Be sure to see the soldiers do their hourly performance at CKS memorial.
(2) “Snake Alley” and Lung-Shan Temple. (These could be seen in the evening or at night.) Snake Alley, as you may have heard, involves snakes being sliced open before your very eyes. Lots of places to eat, not limited to snake!
(3) Danshui, where you can see the old Dutch fort and walk along the waterfront (lots of tourist trap type stuff here).
(4) If it rains, or you’re very interested in Chinese art, then add a half-day at the National Palace Museum.
(5) If the weather is good, and you want some outdoor exercise amidst nature, one possibility is to walk to Zhinan Temple, which is on a hill above Mucha. You have to walk up a flight of 1400 steps to get there, and there are lots of jungle plants around.
Or if you want to walk on level ground, the Guandu bird sanctuary has good views of Yangmingshan and the Danshui river, and there’s an old interesting temple there too.
Other possibilities (involving walking up and down hills) include Yangmingshan itself (being an extinct volcano, it has hot springs / pits of boiling sulfer) or the hills across the river on Guanyin Mountain (where there are several temples–you have to take a ferry from Danshui).
(6) Depending on when you go, it might be possible to catch a performance of Chinese opera or puppet theater.