I was a junior member of my local museum, so if you've something like that near, it could be worth checking. It meant free entry at any time, and it also meant there were various talks and outings to local sites. Had some great days out, and great fun preparing for the annual Christmas competition, which I usually won for my age group, but never the overall prize, and I'm not still bitter about that at all, oh no...
They also helped arrange lifts, if parents didn't drive. (I'm not sure if they did that directly, of if parents sorted it out between them, to be fair. But car-sharing certainly went on.)
In the summer holidays, mum used to take us off to one of the local towns, and we'd go round the local museum and follow a town trail if she could find one. We also went to museums on holiday. (I always like a good industrial museum with old machinery.)
As you can get up to London easily, there are loads of museums there. A lot of the big museums will have "children's trails" and so on. I was at the Museum of London a couple of weeks back. There's also the V&A (loved that when I was first taken as a child,) and the British Museum, of course. The V&A Museum of Childhood is at Bethnal Green, too. The National Portrait Gallery might be good (some Tudor portraits), and the National Gallery, if you look at the history behind some of the paintings. Entry to most is free, except for special exhibitions. There are a lot of smaller museums in London, too, like John Soanes House and so on. And if she likes the Romans, then there's the amphitheatre in the base of the Guildhall, which isn't far from the Museum of London.
Also, lots of places will do special events, especially over the summer, so keep an eye on the various websites. My county council usually has a list of upcoming things for council-run sites (and you may be in the same county.) Also, if you don't drive, check what offers are on at the railway station - they do 2 for 1 offers on a lot of places, if you go by train. (Not sure of the details, as I rarely travel with anyone else.) Hmm, a quick check shows it might just be London, but you can get there, so that's okay. This site about Secret London might also give some ideas - you could look by Tudor or Roman or whatever else.