East Links is good, Almond Valley OK, IME, but personally I don't rate Deep Sea World, I found it quite expensive for the amount of time you spend there. Almond Valley might be better than East Links for a 15 month old and is cheaper, but East Links do have a small indoor soft play if the weather's bad.
There's a soft play called Time Twisters which we prefer to Gambados. It's cheaper and they do a toddler session on Tues, Weds and Thurs where you only pay £3.50 for the session for a little one and the food there is really nice too. Their website is here: https://timetwisters.co.uk/about/
If you don't mind driving north a bit, the Scottish Deer Centre is really good and reasonably priced. I was surprised how much time we were able to spend there tbh! Carnie Fruit Farm which is near the Scottish Deer Centre is really good too. Pick your own and loads of play stuff for kids, plus a huge maize maze. Both are around an hour away from Edinburgh though.
Also an hour away is Blairdrummond Safari Park. Although you might not want to do that and the zoo.
Lots of free stuff to do too, like the Museum, nice beaches and walking in the Pentlands too. The countryside surrounding Edinburgh is absolutely amazing!