Took my 14 month old to the Sea Life aquarium just next to the London Eye. At this time of year it was really quiet so she could happily toddle (and crawl) about. She loved it! It was nice and clean and warm too.
It's quite expensive (about £19 per adult) but if you travel from Clapham Junction get one of the 2 for 1 Days Out vouchers (you'll need to show a valid train ticket at the aquarium).