I'm down to my last three Taiwan posts, finally. For just one week of travel, I sure did a lot!
As far as amusement parks go in Taiwan, the best ones seem to be located far away from Taipei. Like, seriously far away. But given Taiwan's small size, they are all accessible as day trips if you so choose.
The best park near Taipei is Leofoo Village, a 90 minute bus ride from the airport (buses are also available from a couple MRT stations). I arrived in the late afternoon literally just as the rain was stopping.
There are animals in the African Safari portion of the park, but due to the time (the sun was setting) and the weather, I wasn't able to see any of them. Leofoo Village is a park I will definitely revisit should I find myself back in Taiwan, if only for the U-shaped coaster - Leofoo claims it's the only one in Asia, and that there are only two in the entire world.
Leofoo is open from 9-5 or later, though the Safari closes at 4 or 6:30 depending on the time of year. The park has a really good English website though the pictures in the Attractions section don't seem to be working. You can check there for prices - NT$999 for adults, though they have an NT$1299 "Ambitious" ticket too which might include the water park.