Shakespeare's in Park N Shop complex, Jumeirah - they have a little kids room with a tv and games etc
Depending on how old your LO is, Dome in the Jumeira Plaza on Beach Road - right next to Dome is a kids jungle and a separate soft play area which you can see into. :) Those are the only places I'm brave enough to go to at the moment for DD (1yr9months) and that's onnnly if DS (breastfed/2months) is sleeping! :)