Consider staying at the Rosewood for a great traditional experience. If you reserve through a Virtuoso agent (disclosure: I am one, and there are a few others on FT who can also help you), you'll receive full complimentary breakfast for 2 each morning plus a $100 food and beverage credit, which you can use towards dinner one night. If available on arrival, you'll also receive a complimentary room upgrade.

The second night have dinner about a half block away down a somewhat hidden diagonal alley way at The Ships Tavern, a pub that's been there since the mid-1500's. Good fish and chips and excellent lamb. Be sure to eat upstairs. It's as authentic as you'll find in London, the food is good and moderately priced.