Fantasy Island is the best. I stopped going there when I was way too old. (I go to LBI every year with my family and have since I was a baby.) There's also a water park a block or two from Fantasy Island which is perfect for kids that age.
Rent a surrey and ride around the island.
Barry's Do Me a Flavor (https://www.barrysdomeaflavor.com/) has great ice cream and amazing fish tacos. This is down the street from the house we stay in and I probably go there at least once a day.