Just so you know, if beach play is what you have in mind, the Keys may not be the best choice. Thanks to the reef, there really are no real beaches in the Keys, though there are a few little man-made coves here and there. The shoreline of most of the islands is covered by mangrove stands or seagrass flats. Key West may be dominated by nightlife, but in the rest of the Keys visitor activities focus on fishing, boating and diving.
The most kid-friendly destination in the Keys, IMO, is Pennekamp State Park on Key Largo. Unfortunately, toddlers are going to be a bit young for most of the activities offered, which mostly require an ability to understand and follow safety rules -- things like snorkeling, sea kayaking, fishing, and nature hikes. Pennekamp is an absolute wonderland for school-aged kids, though -- nature exploration at its finest.
www.pennekamp.com The only living coral reef in the US is 3 miles off the shore of the park.
If you want warm and beachy, you might try Marco Island, near Ft. Myers. The Gulf water temp today is 75F -- too cold for a native Floridian to want to swim, but if you're the hardy sort you might find it comfortable.
PS: For the poster waiting on passports -- passports will not be needed for travel to Puerto Rico or the US Virgin Islands. Resorts in both of those destinations are offering a lot of specials for this winter to attract US mainland visitors who do not have passports yet.