We are actually not too far from Kawasaki so I decided to check it out. One of the AFWJ mothers was kind enough to tell be about the Sky Court rooftop playground on the 5th floor of Be department store Kawasaki.
The area is very spacious and the floor is covered with that green grass-like felt, so safe for kids to run around. There are many Little Tykes slides around and a few coin-operated rides. Benches are lined up along the side so you can eat your lunch there. And there are drink and ice cream vending machines. If you go with other mothers you can sit and chat while the kids play and rest a bit easy knowing that if you don’t look for a minute your kids can’t run off far. Unfortunately, the place has no roof so bring a sun hat and it’s not a good option for rainy and cold days.
Just next door, also connected to the JR Kawasaki Station is the new LaZona Mall. It’s very new, and there is a wide open air court in the middle of all the stores. 
There are areas to sit and eat lunch during nice weather days. The 3rd floor has a lot of children’s shops including Akachan Honpo.
How to Get There: JR Kawasaki Station
Traveling with a baby? The Kawasaki Station has only an escalator from ground level to the ticket booth level. But inside the Be Department store is an elevator and the store connects to the station. There is no milk room at Be, but there are two in La Zona. One milk room is on the ground level next to the food court and the other is inside the Akachan Honpo store. Both rooms have sterilized hot water for mixing formula.