If you want to stay in Edinburgh then Ather's place is exactly what you're looking for.
It's in a great neighborhood, and only about a 20 minute walk into the city centre. (Though the bus stop is right outside if you're feeling lazy!)
If you have any questions about Edinburgh, Ather's happy to help out. And he's not overbearing either, he'll give you plenty of space to do your own thing.
At one point we even accidentally set off his smoke alarm (sorry again Ather), but he was super chill about the whole thing.
Highly recommended, and great value for money. Frankly, I think he's charging too little.
P.S. His adorable cats seem to love being in the guest room. So I hope you like cats.