Tour London by private car

No, no, no. If you have been to London already, you should know that you cannot simply drive around central London (or any large city in Europe), stop and drop people off, and then wait until they are ready to be picked up again.
Even less can the driver stop at the main sites and get out with you, leaving the car to be towed away.
Parking is very limited, costly, off the street, and at some distance from where you actually want to go. And for most of the day road is slower than the tube.

All you need could be found from any Black Cab i feel.

Sitting in traffic during peak hours (and paying dearly for a private driver) will not minimize travel time. Selecting a good itinerary of sites that are clustered together to minimize distance and using the fastest form of transport (walking, public transit, etc) will come closer to your goal of making "most" of your limited time.

Having a car and driver in London is a complete waste of time and money - there are really no places for a driver to wait for you while you see what you want to see without very significant parking fees, and traffic in London is just a nightmare. If you really want to take cars from place to place then hail black cabs. They are convenient, everywhere, and relatively inexpensive. The drivers are extremely knowledgeable and they will get you where you want to go as quickly as possible since they have intimate knowledge of every square inch of the city.
If you are wanting guides for the different sights in London then hire them at the locations you want to know more about. Most places have guides available who will have deep knowledge of each place.

Suzanne,

I agree with the others that a Black Cab would be a better use of time and money, especially as you stated that you'd like to "make the 'most' of our limited time", want a knowledgeable driver and want to minimize travel time between sites. London Taxi drivers are exceptional as they have to possess THE KNOWLEDGE and the qualities you listed sounded like you were describing a London Cabbie. According to the article I linked, the "test of memorization skills is so intense that it physically alters the brains of those who pass it."

Plan your time by putting closeby attractions together so you are not zigzagging across town then use a combination of tubes and black cabs. The problems of hiring a private car have already been well stated by earlier posts.

The weather will probably be better in October if you have a choice.