Yangmingshan park is one of the most picturesque places within the Taipei city limits. The park is especially beautiful in spring, when its renowned cherry trees yield enormous cherry blossoms. The park is on a mountain slope, and is the location of choice of wealthy Chinese and westerners. The park itself has many trails that offer escape from Taipei's chaos (I guess a charm of Taipei is definitely its ability to offer many such oases). Beware that sometimes the park does get invaded by people seeking the same thing you are, so do try to avoid it on Sundays. But maybe people watching is what you are after, in that case, a slice of life is there for you to see.
You can go to Yangmishan to just hang around and get your bearing back, or you can use a number of facilities offered there. There are hot springs resorts that open late into the evening, and are awfully inviting after a long day hiking. There are thermal areas worth visiting. For those of you interested in a long hike, you can climb Seven Star Mountain, where you can look down one side of the mountain and see Taipei, and the ocean on the opposite side. There are organized camping areas, or you can opt for the more expensive hotels in the area.