I would suggest picking up a car at Heathrow and heading to Stonehenge from there and then onward and northward.
I really like Arnold Clark rental, which is my preferred agency to rent from in the UK. They have locations at both Heathrow and Edinburgh Airport, and are really a delight to work with. I have never had any problems with their service. But you can certainly check prices and compare with other agencies. Enterprise is also recommended.
I was just at Chatsworth last year and absolutely loved it! Make sure you allow plenty of time to see the fabulous rooms and gardens. With regards to lodging, I stayed in Castleton, which I loved and that was about a 30 minute drive, but very reasonably priced. Here is a link to my hotel review: https://community.ricksteves.com/travel-forum/england-reviews/review-of-bull-s-head-pub-hotel-in-castleton-the-peak-district
Another town you might consider is Bakewell, which is about a 10-15 minute drive from Chatsworth. Bakewell is bigger than Castleton and has more lodging options, so you might enjoy that. It would definitely be less expensive than staying on the Chatsworth estate. I checked the price of one of the hotels on the estate isn31c listed and it is a little under £600 for one night. :-)