You could certainly drive it - but I would assume you'd want to do something else apart from driving and sleeping.
I'd fly / take sleeper train to Inverness, hire a car, then drive around Scotland ending up at Edinburgh where you take the sleeper / day train back to London
Have a look at this website which is excellent for working out routes, distances and driving times.
Remember these are minimum times and you will need to build in extra time for meal stops, any sightseeing etc.
London to Edinburgh is about 400 miles and is going to take a full day to drive. Its not the most exciting of drives either until you get well north. So there and back will take out 2 full days from your schedule.
You could do the loop from Edinburgh up the east coast to Thurso along the top and back down the east side through Ullapool to Glasgow in the remaining five days and then back to London. It is however a long time to spend driving and just admiring the scenery through the windscreen.
I agree with Alanrow ... I was all for driving up, but my partner put his foot down and said he wouldn't go unless we flew up ... in the end we flew to Inverness, hired a car there, roamed around northern Scotland, dropped the hire car back to Inverness, then flew back
the trip you've outlined would be better spread over 2 weeks with more breaks up and back ... there's plenty of interesting places to see on the way
-:- Message from TripAdvisor staff -:-
This topic has been closed to new posts due to inactivity. We hope you'll join the conversation by posting to an open topic or starting a new one.
To review the TripAdvisor Forums Posting Guidelines, please follow this link: http://www.tripadvisor.com/pages/forums_posting_guidelines.html
We remove posts that do not follow our posting guidelines, and we reserve the right to remove any post for any reason.