Pardon my inexperience and ignorance, but are you saying that Nagasone Kotestu is the maker of this Kogatana? What then is the meaning of "the 2nd generation, Okimasa?" Is the KEI the small symbol at the end? Is that a Tokugawa Mon? It has been suggested to me that the Wakizashi that this Kogatana came with might be a Tokugawa because of the design on the fuchi kashira, see photos. But, I was skeptical because the design is more of a vine rather than the traditional circle with the three hollyhock leaves that you usually see.
(Is Nagasone sometimes spelled Nakasone?)