There is no question that when traveling in the UK, staying in self-catering units like this one is the best option by far. We did this last year when we were in Cornwall and York and it sure works for us. You can get up when you want, make coffee, have a slow breakfast then face the day.
There seem to be quite a few units available for rent in Edinburgh. We people we decided to rent from have some very nice, centrally located units. When we were first here we stayed on Jeffrey Street, about three doors down from the Royal Mile and a short walk from the Airport bus or the rain station. This time we are in Greyfriars, very close to the University of Edinburgh, the meadows and of course the Greyfriars church.
On top of that I really like the way the people we are renting from handle things. There are lots of options but I would recommend them to anyone wanting to visit this city. They are Edinburgh City Apartments Check out their site.