I haven't read the replies yet, so may be repeating what others have said, but here's my twopennorth:
Late May is a good time. Weather hopefully will be warm but not too hot and evenings will be getting longer. However, depending on exactly when you are there, it may (!) be school half term, so big attractions and family restaurants are likely to be busy. Places like the British Museum and the Natural History Museum will be heaving.
Be prepared to have your bag checked on entry to places. They were doing checks at the National Portrait Gallery last half term. It's just a quick look, you won't be asked to turn it out.
Get comfortable shoes and an A-Z street map or similar and get walking. Yes, there are the big attractions to visit, but so much of the history of London is in the streets and buildings. I love wandering around the City and poking into back streets ad courtyards and looking into the churches. Some of them, such as St Bride's Fleet Street and All Hallows by the Tower, are mini museums.
If you go to the big museums and galleries, don't try to see everything. Pick a particular gallery to focus on.
The Museum of London at Barbican is closed, but the Docklands Museum is well worth a visit if you have the time.
Ignore the tourist buses, take an ordinary red double decker. Keeping London moving - Transport for London is the official website where you can find out all you need to know about bus and Tube routes and tickets.
Similarly, ignore the tourist boats on the river, take an Uber riverbus. Down to Greenwich and back is a good trip.
If using the Tube, stand on the right on escalators, and have your ticket ready as you approach the barrier.
Read Samuel Pepys' Diary, especially 1665-66 on the Plague and the Fire. And this is a blog I like which focuses on more obscure aspects of London history: A London Inheritance - A Private History of a Public City
Have a great time!