Having thought about your question, and having read Ian's answer...
I tend to prefer the A1 route up the east coast but that maybe because I am more used to it.
Some ideas to get you to Edinburgh.
Coming out of London - Hatfield House where Elizabeth I grew up and not on the usual tourist map
I like Ian’s suggestion of Stamford, which is a lovely unspoilt town. It used to be an important staging post but in the 1840s, the town refused to have anything to do with the new fangled railway and since then has rather been bypassed by time.
Moving north, Haddon Hall in Derbyshire is another undiscovered gem
A bit off the A1 but never the less worth visiting is Beverley, the county town of East Yorkshire. It is a thriving market town with a lot of small independent shops and a stunning Minster. This is possibly the best Gothic building in Europe (and unlike nearby York Minster) you may have the building to yourselves.
Durham is a bit more ‘main stream’ but again is worth visiting for its glorious Norman Cathedral and castle (now part of Durham University, but running tours) built high above the River Wear.
Hadrian's Wall certainly fits the description for amazing scenery, particularly if you have time to walk along any of it.
In Northumberland, there is the small town of Alnwick with its castle which is the Home of the Duke and Duchess of Northumberland and award winning gardens, Close by is Warkworth with its ruined castle.
One you cross the border there are the ruined castles of Dirleton and Tantallon on the coast. Alternatively head more inland for the great border abbeys of Jedburgh, Melrose and Dryburgh. If you take this route, think about adding in a visit to Traquair House, the oldest inhabited house in Scotland.
If you want something completely different there is Cairnpapple Hill, to the west of Edinburgh, with a history dating back over 4000 years.
How are you intending to get to Skye - the bridge from Kyle of Lochash, the ferry from Mallaig or the tiny Glenelg Ferry? This decision will decide route and suggestions for off the beaten track.
I've deliberately kept descriptions brief but could wax lyrical about all of the places mentioned...