We stayed in a rental near Stockbridge (a bit closer to the RM-- our address was Jamaica Street) and I enjoyed it... but it was something of a hike to get to the Royal Mile. On the other hand, it was near the trail that runs along the canal, and the Botanical Garden (one of my favorite places we visited), and a branch of Peter's Yard with these amazing cardemom rolls that were possibly my favorite thing I ate all week.
One thing that I couldn't tell from google maps is that there's a slight incline as we walked up from our apartment to Princes Street, and then another steeper incline up to the Royal Mile itself, which could wear one out. But there are also some good bus lines and if you know where the nearest ones are and have a good cheat-sheet of the lines that can take you home it helps a lot.
I think in retrospect I would have preferred to stay on the end of the Royal Mile near Holyroodhouse so I could have more easily reached the nice hikes in the crags and Arthur's Seat. But that was also because I was getting overwhelmed by the gray stone everywhere in the city itself, and kept yearning for green spaces where I could go for a nice hike in the morning (my husband sleeps late).
I didn't do a trip report, but there are photos our trip here if you are interested, with comments on them to explain what's what. Some are from Stockbridge. We were there for a week, taking a day trip into the Highlands with Rabbie's and one on our own to Rosslyn Chapel (another of my highlights!):
https://goo.gl/photos/D8PwNL5CH5odwFuBA
Edited to add: This was also in September, back in 2016.