We have just returned from a lovely week at Villa Mandalay in Kefalonia, it was perfect for what we needed as a party of 8 (5 adults and 3 kids under 10). Plenty of space to shelter from the sun during the day while keeping an eye on the kids who made full use of the private pool. The ice cool aircon was welcome in each bedroom after a day in the sun!
If you are adventurous I would suggest hiring a car as the villa is quite out of the way and that can make grocery shopping a lot easier too, but we managed to get shopping delivered from a local supermarket with the help of the ever brilliant Emma, the villa manager. Emma helped to sort everything for us, including cabs for the wedding we were attending, and we couldn't be more thankful for her help, she's fantastic.
On the subject of shopping, it is quite possible to walk to the minimarket at Spatia (we did this a couple of times), it is an hours round trip but essential for fresh fruit and it is worth going early (opens at 8am).
The Sunset Taverna just a short walk away is great for a Greek Salad at lunchtime, a Souvlaki at dinner or just for a quick ice cream run. I'd read that the people there could be fairly miserable but if you engage with them and talk to them they are lovely people!
Overall a lovely place and we would not hesitate to go back