I occasionally take a bike with me when we go for a long weekend to the inlaws at Crowthorne, which is not too far from your manor.
From there I ride due north to Stokenchurch via Henley Hambleden & Ibstone in the chiltern hills. There are a few 6% as you get into the Hills in the Ibstone CP area. For me its abot 55miles round trip, for you I guess 40 miles? Its pretty much uphill (600ft of ascent) all the way but that's great coming back.
OK, its not the peak district, but its the best I found down there.