What to do on the eastern side of England!

Head east from Ambleside to the coast and Whitby Abbey - fantastic ruin perched on cliff overlooking the North Sea!

If your going to York you could come to Hull. We are the 2017 city of culture with lots to see and do. https://www.hull2017.co.uk/

Like London our museums and gallery are free and we have some beautiful architecture, I might be a bit biased though.

If you are trying to pack all that into two weeks, you won't have any spare time!

From York, you could travel down the A1 and stop off at Rutland Water, a large reservoir. You could take a trip out on the Water on the Rutland Belle or hire bikes and cycle round - that takes c 2.5 hours. The villages there are chocolate box pretty and it's like a mini Cotswolds without all the tourists. The towns of Uppingham, Oakham and Stamford are all worth visiting. In Oakham, visit the Horseshoe Museum (much more interesting than it sounds) and see horseshoes presented by the Queen and some going back hundreds of years. Stamford, a gorgeous Georgian market town is where Middlemarch was filmed. You could easily spend 3 days in this area.

If you can handle more scenic, narrow roads, then head east through the Yorkshire Dales, via Hawes and Askrigg, and Thirsk if you want to see the James Herriott center. If you would enjoy a stately home, then look at Castle Howard and/or Chatsworth House. The center of the country is pretty narrow, so you won't necessarily be far from the route that you took northbound. The recently complete visitor center at Bletchley Park (near Milton Keynes) is also worthwhile.

Of course there are coastal options and Rick tends to ignore all of Norfolk (https://www.ricksteves.com/europe/england/itinerary), but I assume you only have a few days left at this point in the trip.

P.S. I see that Jennifer's recommendation of Stamford also comes with a stately-home option at Burghley.

Stop in Cambridge. Take walks around and visit the Fitzwilliam Museum.

Northumberland's coast and castles
Newcastle upon Tyne
Beamish Museum
Durham
Weardale
Teesdale
North York Moors
Yorkshire Wolds
Beverley
Lincolnshire Wolds
Norfolk Broads
Suffolk

you don't have enough time!

You could look around the Yorkshire Moors - visit Castle Howard, Helmsley and Reivaulx Abbey, (one of my favorite parts of England) or if you want costal villages go towards Whitby and Robin Hoods Bay (North Sea coast). Whitby to Cambridge is about a 4 hour drive. As you head south, you could stop at Ely/Cambridge. Outside Cambridge there is Audley End House. Saffron Walden, Thaxted and the village of Finchingfield in Essex are pretty villages and near the M11. You could easily stay in this area and get to Heathrow the next day.

You have a number of wonderful stately homes in the Peak District, Derbyshire - Chatsworth House, Haddon Hall, Hardwick Hall, Kedleston Hall.
As previously mentioned, Burghley House at Stamford, Lincolnshire, built by Elizabeth I treasurer, William Cecil.
Cathedrals at Beverley, Lincoln, Ely and St Albans, along with the university city of Cambridge.

Another vote for Northumberland and Durham.

If you watch or watched Vera or Inspector George Gently you've seen the beauty of Northumberland. Durham castle and Durham appeared in George Gently. Besides being a World Heritage site Durham Castle was also home to Hogwarts in a Harry Potter film.

I could drive for days in Northumberland and never be bored or dissatisfied with the villages, ruins and vistas.

Detour into Norfolk and go down the coast. There are more pituresque places than you could shake a stick at

I agree with Rebecca: Stop in Cambridge. Take walks around and visit the Fitzwilliam Museum.

My favorite views of Cambridge are seen from the river. Rent a punt from the Anchor or Mill pubs (on or near Silver Street) and head down river past Queens', Kings, Clare, Trinity and St. Johns Colleges. If you are uncertain about punting you can paddle it or hire a punt complete with punter. The advantage of the hired punter is that he/she will not only do the punting but also give you some of the history of Cambridge and the colleges.

Thanks for all of your suggestions! I may add on another week!