There is a good river along highway 28 on the way to the sea, just take the palace museum road and turn right off it...
Also a few on Yangming Shan, if you know where to look -- like up above Yangming village, the ravine that comes out above the satellite dishes has a great river. It is hard to describe how to find it, but you can if you really need to... On the other side of Yangming mountain I found a great "hidden" pool coming down in a creek spouting down towards Danshui; this pool is quite high up, and the water is spouting directly from under the earth just above, so it is really clean; it was last summer, about 35 degrees hot outside, and so I had to clamber down and slip over the rocks until I could dip in... That was a taste of paradise... So cool in the middle of the heat! Wish I'd had a bitch with me then. This summer, I intend to check out some of the creeks and pools way out of the city, like the one on highway 28.
There are two pools in Beitou, one is open year-round and they are outdoors... Only fifty nt or so to get in... Also, at the Beitou train depot there is a huge indoor pool underground in the recreation center, you can get a cheap membership there...