Many thanks Dean.
Firstly welcome to the Corfu Travel Guide. Adding to Dean's comments above, I would also recommend Kontokali as a perfect example of what your are looking for, for your daughter and grandaugher. Kontokali is situated on the east coast, approximately 8km from the airport, 20 to 30 minutes transfer time. Family run hotels I would recommend, is the Spifti Prifti and the Zefiros. Kontokali as a wonderful beach, with shallow water out to a fair distance. The beach is part of the Kontokali Beach Hotel, but is open to other holidaymakers from other hotels. You can use the facilities at the hotel, and the beach also offers a small cafeteria run by a lovely lady originally from Nottingham.
Gouvia, the next resort along, is more livelier, but has a family orientation about it. Suitable for all ages, with many restaurants, tavernas, and shops etc. It is in walking distance from Kontokali on a very flat coastal road, which would take about 20 - 30 minutes to walk. There are a few resting establishments on the way if needed.
Between the two resorts is Gouvia Marina and Beach. This beach is made up of shingle and pebbles, unlike Kontokali Beach which is sand. However, Gouvia beach is just as lovely with its settings inside a coved area.
You also have the Lagoon, which is a large pool complex set in gardens. This is attached to another hotel, but again can be used by other visitors.
There is a very good bus service to and from Corfu Town, if one fancies a good day out shopping and for meals. It is a No7 bus (Blue), costing about 1 Euro each way. The bus stops are situated on the main road to Corfu Town.
If there is anything else you wish to know, do please let me know.