What sorts of things are you and DC interested in and like doing?
The free museums, obviously - the big name ones such as the Science Museum, the V&A, the Natural History Museum and the British Museum. Also the permanent (free) exhibition Treasures of the British Library, and just down the road from it the Wellcome Collection. And close to those are the various UCL museums
www.ucl.ac.uk/culture/
Wandering round Whitehall, Trafalgar Square, St James Park, Buckingham Palace, Green Park, Hyde Park and Kensington Gardens (pop in to Harrods and Harvey Nick?)
You could get the Thames Clipper from Tate to Tate (see all the interesting bits of central London from the water, at ordinary travel rate, not tourist boat prices) and then either wander back along the South Bank or go on towards Tower Bridge and the Tower of London (not free, but big/interesting enough to be worth the entry fee) or get the Clipper on to Greenwich (Cutty Sark, market, Maritime Museum and Royal Observatory)
You can get guidebooks with walking tours of the City, and the Bank of England Museum is worth a visit (free IIRC)