I have a total of over 14 'full' days visiting London and there is still a lot I haven't seen. You can easily spend you entire 10 days in London and only scratch the surface.
If you're into museums, many of the state-owned are free. The British Museum is so large that it would take a week to see and comprehend everything (if you plan to go, develop a plan beforehand).
To get an idea what many/most tourists go to London to see, check out the London Pass web site. (I'm not endorsing the product but it's a good "one-stop" web site for the top tourist attractions.) Figure out what you want to see.
The Buckingham Palace Changing of the Guard can be very frustrating because of the crowds. However, Royal Guard ceremonies also occur, usually with much smaller crowds, at the Horse Guards Parade and at Windsor Castle. There are some very good web sites devoted to everything 'guards'.
Many of the big attractions can be "all-dayers" if you let them. Have a time limit and plan of attack before arriving.
Get familiar with the Transport for London (TFL) web site so you can plan your public transportation itineraries. Once you've decided what you want to see, use the TFL trip planner to figure out how long it will take to travel between sites. BTW - in my opinion, this web site is the #1 reason to purchase a pay-as-you-go SIM card for your smart phone.
10 days in a big city can be fatiguing even when the city is as wonderful as London. You may want to plan a day-trip or two outside the city. Although you can get to most of the countryside via public transportation, I'm actually a fan of the escorted (coach) tours because they 1) take care of the transportation and entrance fees, and 2) add a little structure to the trip. Just Google "London Day Trips." (I use Viator to get an idea of tours available around places I've never been...)
In August, a lot of Europe will also be on Holiday so expect crowds wherever you go. Places in London to escape the crowds are the larger parks (Hyde, Regents, etc.) and the Botanical Gardens at Kew (easily reached by the Tube).
With 10 days in London, I would suggest a holiday apartment over a hotel. My favorite neighborhoods are in the vicinity of the SW7 postal code (you can search London SW7 in Google Maps) for the good transportation options and walkability to major sites.
One last thing, Buckingham Palace is open to the public Late July-September. It's an unforgettable tour. You can find details on their web site.