Oh we were into tons of this days out stuff when I was younger...
Chestnut Centre is brilliant - otters are great, I'd really really recommend it especially at this time of year before the animals go into hibernation.
Tropical World - pretty good day out, and I liked the smaller Butterfly House (it's got an animal farm bit too)too.
One place we went to on a school trip was Blist Hill Victorian Village (but it's in Shropshire...) which is great, though probably not suitable for such a young childlet.
Chatsworth Playground is bloody amazing for kids, though she's a bit young and although there's a smaller area for that age kids, you do have to pay to get in so it's a bit silly to miss out on most of it.
We used to go to York quite a lot when I was little - bloody loved the place, especially the Railway Museum, I remember queueing hours for Jorvik and being a bit disappointed, and quite pricey too. When she's older I'd recommend the York Dungeon, it's great
Urr....Chesterfield Park has a boating lake and miniature train.
If you're willing to go that far, Kinderland in Scarborough and Peasholm Park next door would be perfect - Kinderland is a 'theme' park aimed at young children. On the same note I've heard Gulliver's Kingdom is good for this also.
One thing that also catches the chain of thought.....
Anyone know if Pett's Corner (Sundown) is still open? How amazing would that be! (If it is, you HAVE to take her)