London, Paris, Florence and Rome

I have only two weeks, ...... and i only have approximately 7 to 8 days

Which is it, 2 weeks or 7/8 days? The best thing to count is nights, If you are landing in London on 21st August, that is your first night.
Every time you change location, you loose ½day packing, travelling, checking in and unpacking. Never stay 1 night anywhere as you will have no time to see anything, minimum 2 nights, 3-4 better.
7 nights = 2 cities (4+3 nights), 3 cities (3+2+2) would be too rushed. 2 weeks gives you more cities.

Once I land in London, i would like to leave immediately for Europe

Why? If you want to go to Paris first, fly to Paris, not to London. Otherwise you are spending all your time travelling.

First you need to decide which cities you want to visit. Line them up on a map (no doubling back). Work out how to get from each one to the next (train or fly).
Then that decides where you start and end. Fly "open jaw" into the first city, and out of the last one.

You have 7-8 days to visit Paris, Florence, Cinque Terre and Rome. No way.

The first time I visited Paris, I stayed six days. Rome, seven, Florence 4.

Pick two cities and go with them. Transport is a factor as is time traveling. If going to London, stick to what is near. Paris would work, then you could visit places nearby like Brussels, Bruges and Ghent in Belgium.

Afraid I have to vote with the others, Wendy: 7-8 days isn't nearly enough time for Paris, Florence, Cinque Terre and Rome. Flying into London also doesn't make a lot of sense if you're going to immediately leave for the continent but I suspect you've already booked your flights?

Simply because of the shorter distances between them, I'd probably go with Rome and Florence. Fly into one and out of the other, and fast trains cover ground between Roma Termini and Firenze S.M. Novella in just 90 minutes. Count the amount of time you have between leaving and returning to London in nights versus days? It makes a difference when figuring the amount of time you really have to work with.

Depending on how much that time turns out to be, it's possible to take an easy day trip from Florence to, say, Lucca or Siena for a look at a third Italian location. Lucca is about an hour, 20 minutes by direct train, and Siena is about the same or just slightly longer by bus or train.

Italy in August is hot. It's also humid in some places like Rome.Before you choose Italy, consider if you'll be able to handle a week of traipsing outdoors in the heat and bright sunshine. Paris and Amsterdam are a nice pair.