Kia Ora Sumaiya,
With young children I would find it hard going staying in Auckland for 4 days without a car. There are car rental companies dotted around the CBD. Nearest to the Hilton Hotel are Avis and Hertz along the Hobson Street ridge.
Many places for a day trip. Piha or Muriwai have been suggested, Combine Piha with the Arataki visitors centre in the Waitakere Ranges.
Drive along Tamaki Drive to Kelly Tarlton's for the Antartica Experience and tunnels full of fishlife, followed by a walk along Mission Bay beach and a meal at one of the cafes there.
Devonport by ferry from Pier 1 gives you a foreshore walk to North Head and the navy museum.
Cornwall Park is a working farm, so too is Ambury regional park.
Skytower is always interesting including when it is raining. Weta workshop Unleashed is on Federal Street for a rainy day activity.
The Auckland museum located in Auckland Domain gives you an under-cover Maori concert, and fantastic exhibits including hands-on displays in the kids zone.
Near the Hilton Hotel at the entrance to the Viaduct harbour is the Auckland Maritime/Voyager museum. Auckland has long been known for its sailing vessels, and there are some interesting exhibits. Again an under-cover rainy day activity.