Trafalgar Square, Buckingham Palace, 10 Downing Street, the London Eye, Houses of Parliament, the War Rooms - these are all in a fairly small area and you can easily walk to each one with no need for buses or trains. You can do all that lot in a day if you start early and have enough energy.
The British Museum is good, as are the Natural History and Science museums.
As you'll be here in June, when it should be warmer, then a trip to Kew Gardens, and a visit to Richmond are recommended. You can escape the hustle and bustle of central London. That's another day taken care of.
Oh and Hampton Court Palace is beautiful. And they have a maze.
Fully agree with David S.
Also I think you're right to spend more time in London than in Paris, as it is a much bigger and more interesting city IMHO.
By all means do the 'touristy things', but don't spend the whole time waiting in line for the likes of Mdm Tussaud's (a bit of a tourist rip-off, if you ask me), take time also to just walk around, sit in the park and watch people go by, etc., to get a feel of the actual cities themselves.
The Harry Potter Tour is outside London and really only worth travelling to unless you are a fan. What you do and see really rather depends on your interests. Take a look at www.timeout.com/london
For your journey from London to Paris take the Eurostar train. It takes just over 2 hours city centre to city
centre and is therefore far more convenient and interesting than flying www.eurostar.com
The Harry Potter tour is brilliant even if you're not that into it and Madam Tussauds is amazing they have wax works of basically everyone famous like the queen,one direction etc, Also I don't know if they still do this but when I went we went on a Jack The Ripper tour that was interesting. Also see the sights usual tourists go Big Ben,Buckingham Palace, Houses of Parliment, The London Eye and the Olympic Stadium.
London - these links get you all the shopping you may want to explore here in London.
https://www.google.co.uk/#q=street+sensa...
This link gets you all the latest on what's happening here in London
http://www.allinlondon.co.uk/
How to get from London to Paris - by Eurostar from St. Pancras International London to Gare du Nord Paris.
https://www.google.co.uk/#q=eurostar
Eurostar via Youtube
https://www.youtube.com/results?search_query=eurostar
Your Eurostar Train will depart London at X...time and you will be racing across the Pays de Calais France in one hour - and in another one hour and fifteen minutes you will be arriving at the Gare du Nord station in Paris.
Saint Pancras International London
https://www.google.co.uk/webhp?tab=ww&ei=_heqVMOCHsK4aeu4gZgH&ved=0CAYQ1S4#q=saint+pancras+international
Gare du Nord Paris
https://www.google.co.uk/webhp?tab=ww&ei=7BiqVJWPO8ftaN2ugsAC&ved=0CAYQ1S4#q=gare+du+nord+paris
Accordion Paris via Youtube
https://www.youtube.com/results?q=accordion+paris
While in Paris don't forget to pay this gentleman a visit. The Thomas Jefferson statue in Paris. He is facing his favourite hotel next to the river.
https://www.google.co.uk/search?q=thomas+jefferson+statue+paris&biw=1600&bih=791&source=lnms&tbm=isch&sa=X&ei=8RmqVN7dPMfbapjlgMAD&ved=0CAYQ_AUoAQ#facrc=_&imgdii=_&imgrc=ibD5xqip1ywO8M%253A%3BOs-KSyNINRCn3M%3Bhttp%253A%252F%252Fparis1900.lartnouveau.com%252Fparis00%252Fstatues_de_rues1%252Fpage3%252Fstat_tjef.JPG%3Bhttp%253A%252F%252Fwww.armchairgeneral.com%252Fforums%252Fshowthread.php%253Ft%253D72079%3B518%3B700
bon voyage
It's etc., not ect.
