I was staying at the same hotel last year. Tried quite a few places nearby for breakfast including the highly-acclaimed & over-priced The Wolseley (160 Piccadilly, about 15 minutes' walk away).
The best breakfast I had in all two weeks I was there was actually at the Hard Rock Cafe London (!) - maybe it was because their toast & waffles came out perfect that morning.
But for the quintessential British breakfast experience, Dorchester Grill Room is great, although I'm a loyal fan of Simpson's-in-the-Strand (you'll need to take a short cab ride there).
Inn the Park (at St James Park) is also quite nice and the Fifth Floor at Harvey Nichols (they both opened from 8am on weekdays).