She'll love Jacob Ballas - it's open to the public - but check the days - I think it may be closed on Mondays. Just make sure you slather on the sunscreen & wear a hat - there's not a lot of shade. They also have a small water play area so if she likes the water bring a swim nappy & towel.
There is a tiny indoor playground at Alexandra road you can check out the info here (also closed on Mondays):
http://www.playtime-recreations.com/
There is a small playground in the Forum shopping mall that is perfect for toddlers. It's on the 2nd floor near customer service. You need to buy a $10 membership for 2 years but definitely worth it if she is planning on coming back to visit.
The place across from Tanglin Mall is the Tanglin Tree restaurant. Great place to have a bite and there is a little playground.
At 15 months she'll probably just like being pushed around - and going to the park. If you're at Douby Ghaut - just take a stroll down to Paragon as well - there is another playground on the 5th floor, a smaller one around the back of the 5th floor by the VitaKids shops and also there is ELC where she can play with the toys.