Rent a car in London to drive to the Lake District? Crikey - No! That's how to ruin the trip within minutes.
Use the train to cover long distance and avoid the very crowded motorway network - it's congested all the way from London until about Preston.
You can rent from the big name major rental companies in Lancaster or Carlisle. They're nearer the station in Carlisle, but Carlisle does mean going about 20 miles too far then backtracking.
Either return the car to where you picked it up - or a one way rental to Edinburgh (avoid the city - drop at the airport) will take you through very pleasant Borders scenery. No traffic worries until you reach Edinburgh.
You need a licence that is more than a year old and to be above 25 - expensive age based supplements for young drivers. Automatic transmission needs to be specifically orders - rental fleets are mostly manual. Damage control is far stricter than North American visitors are used to.