As you probably know, London is a super-expensive city for everything, but there are places you can go for under $20. Laughing Halibut is one of my favorites. Great fish and chips, although it looks a bit sketchy from the outside. I've made many a trek to this place for their cod. I would also try the Regency Cafe which is a phenomenal breakfast diner. You may have to wait a while to get a seat, but it's well worth it. If you want to go a little more upscale without spending a lot, you might want to try the Bluebird cafe. It's basically a cafe attached to one of the hippest stores in London. The food is good--not great, but that's not what you go there for anyway. Ye Olde Cheshire Cheese is not only one of my favorite pubs in London, but one of my favorite restaurants. They make some kind of mushroom casserole thing that is fabulous even if it sounds horrible. Tayyabs is wonderful and cheap for Pakistani food. Although beware--it is literally the hottest, spiciest food I've ever tried in my life--I was literally sweating as I ate it. Huge lines, too, so go early. Leon is a restaurant chain that's very popular, although my feelings about it are mixed. I thought it was OK--not incredible, but cheap and there are a lot of locations. If you want to feel as if you're splurging without actually splurging, you might want to try the oyster bar at Bibendum Oyster Bar where you can pay by the oyster. It's an amazing converted space that used to be owned by Michelin (thus, all the Michelin logos and symbols all over the place). It's one of my favorite spots in the city for people watching and in a lovely neighborhood that's fun to stroll through afterwards. I know this sounds a bit touristy, but try the food halls at Selfridges. They're less expensive than the food halls at Harrods and I think the food is actually better. You'd have to order sort of strategically and maybe only go for breakfast, but I would also recommend The Wolseley. It's become a bit over-trendy, but it's still a great setting with great food and a ton of options. Again, you'd probably only be able to do breakfast here, but The Botanist is one of my most favorite restaurants in town. I love the atmosphere and the food is always reliable. It can be a little pretentious, but whatever. 

Try  Wagamama and a couple of the restaurants in Wardour Street in Soho. There is a mexican place there that is very cheap and cheerful but can't remember the name now. 

If you are on a really tight budget try Kebab shops which serve up Burgers,Doner kebab, Fish n Chips, and more...  If you prefer the tastes of the Orient there are many budget friendly  Chinese and Vietnamese restaurants scattered throughout  London. Also, if you don't mind eating in a pub most offer great food on a budget.

In london, for that money, is best to buy in a supermarket (Tesco, for example) and eat in the hotel.