We are staying at the Hilton in Slussen for 3 nights before our July 31 cruise. We were able to get it for points and it is less than 2 blocks to T-bana/public transit station and walking distance to Gamla Stan. However, would likely take T-bana to the Gamla Stan station just to cut down on walking. 8 minute walk from ferry dock.
So, we are not far from where you stayed. Do you remember approximately how much the taxi ride was? It would be so much more convenient from airport to hotel directly.