Marchmont and Bruntsfield near the Meadows is a pleasant walk to Old Town.
Alternatively you could go the New Town side and stay in Stockbridge.
I lived there 6 years ago and would live there again and these areas would be where I would stay. I'd also stay down Leith. about 7 years ago
I lived there ten years ago so things may have changed but areas on the Royal Mile, in the Grassmarket and further afield like Duddingston Village have maximum atmospheric impact and I would recommend them for the full 'Edinburgh experience'. They will also be close to most of the tourist action, so minimal walking. about 7 years ago