Your budget is right around 20 pounds per day. That should cover you for food and a few extras, as long as you're careful. You won't be able to eat 3 meals a day in restaurants - not even cheap ones. But if you focus on getting most of your meals as cheaply as possible, you'll have money left over for the occasional splurge.
With a kitchen, this is a lot easier. Even if you don't want to do proper cooking, grocery stores sell all kinds of convenience foods that you can use for cheap meals. I think British "ready meals" taste a lot better than US TV dinners, but opinions vary on this. There are also standard staples like sandwich fixings, cereal and milk, etc.
Fresh fruit can be an excellent bargain in London. You might want to look beyond the grocery stores to market stalls, greengrocers, and produce stands. Some have bargains like a giant bowl of oranges for a pound.
When you're out and about and need a meal, inexpensive options include bakery sandwiches, Cornish pasties, and the like. You'll find places all over that sell prepackaged sandwiches for just a couple of pounds - these can be kind of gross, and I get heartburn when I eat them every day, but they're cheap, portable, and filling.
That's just for food, though. You don't say what kind of "places" you'll also be spending your money on. Some London attractions are very expensive and can wipe out your whole budget for the day. On the other hand, many of the best museums are completely free. So choose carefully.