There are so so many things to do in London - what is she interested in?
Off the top of my head - museums such as the Natural History Museum (dinosaurs) are free. The Transport Museum in Covent Garden is not free but it is great of that's of interest.
A ride on the top deck of a bus is a great way to see lots of sights - St Paul's and the city skyscrapers are quite a sight, and the bus is basically free (free for DD and just part of the public transport spend for your mum).
Mine at a similar age (maybe a bit bigger) enjoyed a walk around Trafalgar Square (lions and Nelson’s column) and then down to the river to look at the H of P and Big Ben; IIRC we walked back via The Monument which commemorates the Great Fire of London, which they had learned about at school.
Hamley's toy shop is great - they have brilliant displays and things to look at and then you end up buying something lol.
The Diana Memorial Playground at Kensington Palace is great too and free as well.
I would suggest tho you just plan a couple of things as it can be quite exhausting. Make sure there is time for a nice stop for lunch.
Edit: Wow just checked out the Paddington experience - it would cost £50 for tickets for your mum and your DD! Lots of great stuff in London for less than that.