Before we moved here we stayed at The Stafford during our trips to London, and I remember always being underwhelmed by the food in the area, and with the recommendations given to us by the concierge.
Now having lived here and indulged greatly in the food scene, I'd go to:
1. Koya. Very casual, no reservations, and you'll probably have to wait in line, but the food is incredible.
2. Nahm. David Halkin's Thai temple in the Halkin Hotel.
3. Amaya. Upscale, trendy restaurant with fantastic Indian food.
4. Ceccione's. Italian in Mayfair. I've had three very good dinners there.
5. Zuma. On the trendier and louder side than I like, but it's never disappointed me. Unlike it's sister restaurant Roka, where I've had horrible food.
6. Yashin. New-ish sushi restaurant with "experimental" omakase. Very, very good.
None of these are overly far from where you'll be staying (and try to get put in the new section of the hotel).
The Stafford is excellent, and has a very good bar and kitchen.
There are many good places within 10 minutes walk (the Ritz hotel's dining room is gorgeous).
For more casual places, these two are good, and in nice locations. About 20 minutes walk or 5 minutes and £10 in a cab.