Another vote for the Covent Garden area. We tend to stay at the Strand Palace which is on the Strand (surprise, surprise

) and within 5 minutes walk of Covent Garden and Trafalgar Square. Leicester Square is about 15 minutes walk away and so is Picadilly Circus. The hotel is reasonably inexpensive for London and is of a good standard. I know lots of American folks stay there. They have a website here
http://www.strandpalacehotel.co.uk/
As everyone has said you'll have plenty to do. One of my favourites is the Natural History Museum which is great for kids (and big kids) particularly if you like dinosaurs of the large and snarling variety.
I also love the British Musuem which has some really great stuff like the Rosetta Stone and the Elgin Marbles. The only problem with the museum is that it is enormous.
The Tate Modern gallery is in a building that used to be power station and it's worth a trip just to see the Turbine Hall which is a huge space. The gallery is full of strange, odd, freaky but great stuff (think Salvadore Dali, Rothko and Bacon

).
You might also like the Cabinet War rooms and Churchill Museum which is very interesting for adults but you might have to bribe the kiddie winks with a promise of a trip to Hamleys Toy Store.
If you want to see a show you can't beat the Lion King and the theatre is right next door to the Strand Palace (I think of everything!!). If you do want to see the Lion King try to get aisle seats in the stalls and sit your two daughters next to the aisle. When the show starts and Rafiki starts singing you need to look behind you - it makes me cry every time
One tip is to avoid Oxford Street - it is very busy and since it is a big shopping street it does attract pickpockets. People can also be quite impatient which is not fun for little ones . If you want to shop, Regent Street, Covent Garden and Kensington High Street are all much nicer although a little more expensive. Covent Garden is mostly pedestrian so no cars and there is usually lots of street entertainment (singers, magicians and the like).
Food ranges from the inexpensive (McDonalds) to the hugely expensive (anything run by Gordon Ramsey) but there is plenty in between. You will tend to find food more expensive than home however. What we tend to do is eat a large breakfast then have lunch out and a lighter dinner (food is generally more expensive in the evenings when the restaurants want to catch the theatre goers). There are sandwich shops and coffee bars everywhere you can always get something light and relatively inexpensive. There is also a Rainforest Cafe near Picadilly Circus which always reminds me of WDW.
You'll have a great time I'm sure.
