I think the recommendation to find a hotel in Kenner is a good one. Audubon Park and Zoo is an obvious choice for toddlers. The Zoo did not take extensive Katrina damage and my boys loved it at that age. City Park had extensive damage and flooding and they are only now getting the damage fixed. Aquarium of the Americas is good for kids that age. I haven’t been to the Children’s Museum so can’t comment on that.
With little ones you want sights and sounds. If you are in the French Quarter around 10 or 11 (you’ld have to check the times), the steamboat Natchez leaves upriver from Jackson Square. You can watch the spectacle from the Moon Walk across the levee from Jackson Square. The cathedral bells toll the hour and the Natchez has a steam calliope which is played. Actually, just sitting on the benches at the Moon Walk is a nice activity. You can see the ferries crossing the river and the ships passing up and down the river. (They’re just too young for this but a cheap thrill is taking the ferry across the river to Algiers and back).
Cafe du Monde - without a doubt. The kids can have hot chocolate and beignets. Show’em how to dunk. I’m not certain if the French Market has fully reopened or not but the sights and smells are memorable.
The St. Charles streetcar line is not fully reopened but it is partially operating. Nice trip for little ones.
Blaine Kern’s Mardi Gras World across the river in Algiers. Mardi Gras floats…
If they were a little older I could recommend far more but this is pretty much what I took my two to when they were that age.