Cheap things to do in London

DullGreyGuy

Science Museum and Natural History Museum are the two obvious ones with most the exhibits being free but there are other options like the Royal Institute of Science and its Faraday Museum, they also do talks/workshops but they tend to be paid for (but often relatively modest fees). 

Can see other options https://londonplanner.com/science-days-out-in-london/ 

Eldi_Dos

The cheapest thing to do in London is go for a walk.
From hotel make your way down to Marble Arch,the most boring part of the journey,cross over into Hyde Park.Make your way via Lovers Walk to Green Park and cross onto The Mall then up to Trafalgar Sq.
A stroll down Whitehall will bring you to Big Ben where you can join The Embankment to the Millennium Bridge,turn off there up to St Pauls and onto Liverpool St Station.
People come from all over the world to see the sites on that walk,and it is all free to you on the walk to the station.

Timalay

Update, money wasn't as tight as I thought it was going to be coming towards my pay date on the 19th, and I have booked the Frameless Immersive Art Experience near Marble Arch.  As a big bonus, it's only 15 minute walk from the Hotel.