Upstairs at the Al Muhairy Centre has a great play area - it's very cheap too! There's a baby/toddler area and softplay/climbing for slightly older kids. My top tip is to buy a coffee from the cafe on the ground floor and take it upstairs where there are tables and chairs for you to sit at whilst watching the kids play!
i like My Playground in Marks and Spencer. It's clean with areas suitable for young ages, but u (or ur nanny) should keep a close eye on the baby since some unsupervised kids go crazy sometimes.