I’m chiming in a bit late, but if it’s not all wrapped up yet...
A tip is to book everything in advance, including free museums—otherwise, you’ll face an insane queue and risk not getting in if it’s full.
I’d add to the sightseeing suggestions: the Lego store in Piccadilly Circus, which will interest even your teen!
Right across from it, the M&M’s shop (with outrageous prices!).
With a teen, you *have* to check out Camden Town.
You can get there by taking a little tourist boat ride from Little Venice. Nothing exceptional, but it’s original and not too expensive.
For accommodation, we stayed in London Stratford (Aparthotel Adagio—pretty pricey, but the amazing breakfast lets you get by with just a sandwich at lunch, and you can cook in the evening since there’s a supermarket nearby). It also has the advantage of being close to the Tube and on the line to Stansted Airport if you’re flying with a low-cost airline.
For a slightly gentler price (though still...), the Ibis Style London Southpark is very well located.