I have family in Asheville, and we head out there fairly regularly. There's a lot of hiking and outdoorsy-stuff, but at 2, DD is just too young to really enjoy it for more than an hour or so.
I'd recommend:
Health Adventure (downtown at Pack Place) - dorky name, but a really nice kid's museum. It's in the same building as the art museum, which I wouldn't recommend for kids that age.
Western North Carolina Nature Center - sort of a mini-zoo with locally indigious animals. The exhibits are close to each other, and you get a good view of the animals generally. They have some hands-on, play with the turtle shells/fur/etc displays too.
http://www.ashevilleparks.org/parks/parks&play_areas.htm has a listing of parks in Asheville, and some of them are very nice.
Bel Chere is loads of fun (and has lots of kid's activities) but that's the week after next.
They might like shindig on the green, if they like music. My kid would happily bop away for quite awhile.
Fun Depot is nothing really exciting, but a good way for little ones to run off some steam in an indoor play area. Your daughter could do go-carts with an adult driving, if she were so inclined.
I've been tempted to do the great smoky mountain railroad, but my kid isn't a train fan, and it's a 4 hour trip. For train lovers it would be loads of fun though.
Rent a boat on Lake Lure? Or just go on a guided cruise? There's a beach at Lake Lure that rents innertubes and has bumper boats, or just a chance to feel the sand in your toes.
Tweetsie Railroad could be fun, but it's kind of a long drive. That's a western/railroad-themed amusement park aimed at little ones.