Ciao Susan,
What have you looked at already or who have you contacted? Since I have no way of knowing about dates, I can recommend places you might already have contacted and heard back that they don't have those dates free.
Also how far of a drive from Florence are you looking at doing? Would the area south of Siena be alright? It's a further drive back up to Florence but you can take the bigger roadways so it isn't a long drive.
Just know that private villas for just 6 people are very rare. Villas are huge so tend to be for larger groups, so many of them have been divided up and offer accommodation in portions of the villa called apartments since they now have their own kitchen, living area and outdoor space, even if the pool will be shared among the various apartments. Here is where you can find a variety in sizes of villas and estates - there are those that have 20 apartments and those that might have 4 or 6, so you can see there would be a difference in the overall effect of sharing a pool and other common areas.
For example, off the top of my head, have you already contacted Borgo della Meliana or Romitorio di Serelle or Agriturismo Incrociata? Incrociata is on the smaller side in terms of the estate/number of apartments. These are to the south of Florence, but you can also consider Fattoria il Lago and Dianella, a wine resort that offers suites instead of apartments, but has its own restaurant, so there are benefits to that.
Let me know about places you've already looked at and I'll try to offer some more places for you to consider.