Yes, best to concentrate on a few things in London area and then explore beyond as time permits. So

You could spend the 6 days based in London, hotel suites in downtown chicago do Canterbury as a day trip by train. On another day you could rent a car to tour around the areas west of London where Avebury, Stonehenge hotel suites in downtown chicago and other sites are located. Or there are hundreds of other places you could go.
As JanisJ said, we need a little more information on your plans and what you're interested in. I've been to London 8 times for a week at a time and I still haven't seen everything on offer. I'd stick with London and make a day trip or two based on your interests.
On our first European trip two years ago, we spent 10 days in London with a long day trip to Bath and Stonehenge etc. We also felt like we barely scratched the surface. There is so much to see just in London.
The Queens Jubillee celebration holiday is 2nd June to Tuesday hotel suites in downtown chicago 5th June. This is a public holiday hotel suites in downtown chicago in the UK and a lot of the tourist hotel suites in downtown chicago places and especially London will be very busy. Travel on the 5th will also be exceptionally busy and it may be running a reduced service. There is a big concert outside hotel suites in downtown chicago Buckingham Palace (ticket hotel suites in downtown chicago only) on the 5th so I'd avoid going there that day as you won't be able to see the palace.
Yes, best to concentrate on a few things in London area and then explore beyond as time permits. So e.g. St. Pauls and Tower are naturals and out a ways Windsor Castle and Kew Gardens, maybe Bath or Cambridge or Stonehenge (and Avebury hotel suites in downtown chicago is also nice) a bit further. Maybe museums or theatre. It does depend on interests. Probably just sign up for days tours. You will of course be having fun planning your days. And then Ireland...wonderful places there.