We are staying at WoWo in Sussex this year that has been highly recommended - although not been ourselves yet.
We have also stayed at Heaven near Haywards Heath which is a lovely farm site with a nature trail and tea room - lots of family attractions nearby.
We stayed on one near the Jack n Jill windmills in Sussex - fairly basic but nice views and again close to lots of family things.
We have also camped near Corfe Castle a few times which is probably getting near your max journey time but doable in a weekend. It is lovely down there. castle itself and village of Corfe Castle lovely. Swanage lovely family resort and lovely steam railway. Also Poole doable and lovely Jurassic Coast.
We stayed at
www.burnbake.com/ a couple of times - lots of families but you can't book so you take your chances although we managed to find a large enough pitch for 4 families last July. They have a shop where you can get fresh bread and a Pizza yurt which also does cooked breakfasts.
We normally choose fairly basic facilities but like somewhere we can have a fire going.