It's the official NBTHK museum, actually
Just to set things straight, this Japanese Sword museum tucked into a corner of a residential neighborhood in Shibuya is actually the official museum of the Nihon Bijutsu Token Hozon Kai (Japanese sword preservation society) office. The place is about 1000 sq ft, and had about 50 or so blades on display when I visited.
The target audience for this museum would be traditionally made Japanese sword collectors. Thus, the exhibits are very high end swords from various schools and swordsmiths. They have limited exhibits on sword mounts, but there is still a great selection of tsubas and menuki. This is the best place to see a large choice of Tokubetsu-Juyo and above swords, in mint condition, with excellent lightning to enjoy the hamon and jitetsu. Since it's usually pretty empty, you can take all day to admire the blades. Sadly no photography is allowed.
I would agree that the Japanese-only text on the exhibits does inhibit non-sword collectors from better appreciating the blades.
I would say that for ~US$6 admission, you get the most bang for the buck in Japan. The place is also hard to find, and I walked from Shibuya station, it took about 20 minutes, and I had a heck of a time finding the non-descript building. I had to ask a lady who was out walking her dogs, and she was most helpful even with my limited conversational Japanese.
For sword collectors, don't forget that the Aoi Art store is literally 2 doors down from the museum, so it's a great chance to see more blades, and maybe even buy some of them! I still have the great pamphlets from the swords on exhibit when I visited, and I love seeing the jaws on my fellow sword drop when they see what was on display.
The Tokyo National Museum in Ueno also had about 8 to 10 blades on display, from some very famous smiths like Masamune, Sanjo Munechika, etc., and you photography is TOLERATED there, but officially not allowed.