If you are looking for a great place to stay that is within walking distance of the old town (10 min approx) then I can only recommend you stay at Kristina's!
Located in Ostermalms-torg, less than 2 min away from the subway is a very spacious old flat uniquely decorated that will take you back in time. The room itself is spacious, off a quiet street and bathroom, kitchen and room are all very clean. Worth noting for those that are sensitive to light that the bedroom does not have black out curtains.
Although my mother and I did not get a chance to meet Kristina in person for she was travelling, Kristina was very responsive via email and check-in and check out was no problem whatsoever.
Kristina had very kindly left some food and drinks in the fridge for us to enjoy a breakfast or afternoon snack. And close-by within a short 5 min walk are 2 supermarkets and a few restaurants, a few of which are Japanese if you fancy a take-away night-in.