The dock adaptor with the Nano is for future accessories, not current ones. All it does is provides an exact fit for the Nano. The Nano will sit fine in any old dock, as the connector alone is enough to support the tiny Nano iPod.
However as the other poster said, the Nano is USB only. Use the included cable with the Nano on the dock you currently have.
Apple has more info on the dock adapter
here. Keep in mind they say the Nano is not compatible with older docks, but this is only from a size perspective, and the firewire issue.