Sorry to say that I don't have much good news for you regarding the use of pools and hot tubs. According to the daily program we received on Infinity this past February, kids must be at least 3 years old and past the diaper stage to use the swimming pools and jacuzzis. The adults-only T-pool is the only covered pool on the ship.
There may be a splash pool just outside the kids center for their use. I'll have to check on that.