I love the North York Moors, the moors are not at all like the dales, and to my mind both are more impressive than the cotswolds, in particular if your there at a busy time. The derbyshire dales (white peak in the south and dark peak in the north, Yorkshire dales and North York moors all have different and unique characters.
Blenheim and Chatsworth are both half-whole day visits, if you get a overseas membership for english heritage you can see several smaller properties each day, and use the stops to break your drives up. The national trust is also a possibilty, but their short memberships are more expensive
some locations to consider: derbyshire peaks, ILAM, Bakewell, Eyam (plague vilage) Castleton is a good route north. from Castleton go through winnatts pass, then over mam tor to Edale. the road goes high enough that you can stop and climb to the top for stunning views in 20 minutes!
From there you can take snake pass (to glossop) then over Holme moss to Holmfirth (now we see the dark peak, much more desolate), then to Haworth (for the Bronte parsonage), then you have to choose either the Dales or the North York moors, but if it the moors, then stay in Helmsley (I think this is the nicest town in England) which has 2 fabulous abbeys (rievaulx and Byland) plus the castle within a short drive.
Where is your starting point? would you consider a different place to stay on each night? in 6 days you can plot a route which doesnt finish a huge distance from the start, but not retracing your steps saves a lot of car time. For me the MOST impressive of all in England in the Nothumbrian castles route, (you could drive along hadrians wall if you are in the lakes, then through Alnwick, Bamburgh and Lindisfarne, after Berwick the scenery becomes less impressive until your on final approach to Edinburgh
