I was there for a week a couple of years ago. I would have needed two weeks more to see and do everything I wanted to.
What kind of nerd are you? Tech? Science? Pop culture? Militaria? History?
In summer there are lots of walks with various themes, I dunno about now when it is off-season.
Do find which museums you can't visit on the Monday and choose something else to do that day.
One tip for visiting (any European city) is to plan strolls through the city with sights along the way. I had one north-south through the city centre and one sort-of along the Themes.
Do get an Oyster card and learn the basic structure of the Tube and bus lines in advance, and it also works on some boats along the Themes.