Tate Modern is across the river and a bit further back towards town - that’s free.
Can you afford to take the bus or walk along to Trafalgar Square? It’s basically one straight line along Ludgate Hill, Fleet Street and the Strand to Trafalgar Square where you have the National Gallery, which is free.
The British Museum, Victoria & Albert, Science and Natural History Museums are all free and excellent. You can reach them all by bus, which is free.
I really wouldn’t go to London and just stay by Tower Hill; there’s not a lot there unless you’re going to the Tower of London (brilliant but expensive) or Tower Bridge or HMS Belfast.
If you’re not sure about bus routes download the City Mapper app or look them up on TfL.
Wandering along the South Bank or going to Borough Market near London Bridge on Saturday morning, or the Museum of London are also great and free.
St Paul’s Cathedral is near the Tower and is spectacular, though you may have to pay if you’re not going to a service.
Hyde Park and Kensington Gardens are also great and can be reached on a couple of buses or the Tube.
For good restaurants I’d suggest going near Borough Market or the South Bank near the Royal Festival Hall. The really good ones are in Soho and other parts of West and North London, but may be beyond your budget.
Do walk to Soho though- it’s near the National Gallery and China Town always looks great; you can eat a cheap Chinese meal there too. I hope you have a great time.