It's hard to imagine a better location for a stay in Stockholm - an apartment in the royal palace maybe? The Nobel Museum is a stone's throw away (alright, you'd have to be a pretty good stone thrower). The ferries for Djurgarden (ABBA and Vasa museums) depart nearby and it's a short stroll to the Norrmalm shops. It's also well-placed for the pleasure boats out into the Archipelago.
The apartment itself is on a quiet side street just off Osterlanggatan - essentially the main thoroughfare of Gamla Stan. It's up three flights of winding stairs (but you wanted the Old Town, right?). The kitchenette is compact, but fine unless you were planning on throwing a dinner party. Do mind your head on the cooker hood. Otherwise, the apartment is spacious and airy, and stylish and comfortable.
TIP: should you decide to walk with your luggage from the Metro, take the long way round rather than going straight across Gamla Stan. You will avoid having the cobbles destroy your wheelie-case.