We had a great stay at Elizabeth's. At first, you notice the poor state of the building's interior and you start wandering about what you got yourself into, but the apartment is nice (if a little small) and perfect for your needs.
Then you realise where you are. Right in the heart of Edinburgh. Within walking distance of everything you might want to see. You can go to the castle in the morning, come back home for lunch and spend the afternoon strolling along the lovely Prince's St. It's the perfect location. Plus, you can get there from the airport with the amazingly efficient airlink 100 bus in almost the same time you would take with a cab/uber at a fraction of the money. Brilliant.