Have just returned from Lassi - my 4th visit. Cannot recommend highly enough and very suitable for your 7yr old daughter.
I have not been in the school holiday period since 1998 but my son had lots of company then.
My first visit in 1989 was quieter than it is now - not so many restaurants and tavernas, but in my view it has grown very little in comparison to other european destinations.
kefalonia as a whole is a beautiful island - you will have a wonderful time!
I cannot help with Liberatos apartments but there must be lots of info on the internet!
We have, yesterday, booked to return next summer!
Let us know how you got on on your return!