The Beitou Parent-Child Center is awesome - and free! If you take the MRT, as you come out of the station, head to your right (towards the mountain). Look diagonally to the right and you'll see a sort of goldish high-rise apartment building maybe a hundred meters away (it's the only one on that side of the road nearby). Head to that and the entrance to the Parent-Child Center is in the lobby.
Warning: they are closed on Mondays, and have a break from 12:00 til I think 2:00. Get there early (opening time is 9:30) because sometimes they fill up even on weekdays, but pretty much always on weekends. Or you can call to make a reservation: 02 2898 3217
Not easy to get through, though.
There are a bunch of other free children's centers like this around the city now:Taipei to get new daycare centersTaipei opens center for children to ease parenting burdens