OK - it looks to be in excellent condition. I don't think there is any doubt it is an authentic, vintage, WW2 blade from Yasukuni smith Yasunori. If the letter you have doesn't specify anything about the person who turned in the sword, then I think discovering who the previous (Japanese) owner was will be a lost cause. I wouldn't fret over this too much. WW2 blades are still problematic to import to, and register in Japan. After the war, the Americans deemed all weapons of any kind to be illegal, so all swords (even heirlooms and museum pieces) were confiscated, and it was only after several months of negotiations did the Americans allow an exemption for family heirlooms and artistic treasures.
Blades made for/during the war were deemed weapons of no artistic merit, so WW2 blades continued to be confiscated, destroyed, or in some cases (like your sword) allowed to be taken back to America as war trophies. To this day, WW2 blades are seen as weapons, and may not be registered in Japan. In other words, its illegal to own them in Japan. However, the old rules are slowly being reinterpreted, and the current generation of Japanese have different attitudes compared to the immediate postwar generation, and also there is no occupying army watching over the registration process, so we are starting to see a bit more leniency in regards to the registration of some WW2 blades (including Yasukuni blades). Anyway, it's a grey area and if you or the import agent isn't well-versed in bringing in a wartime blade, it could end up being a big headache for everyone. Plus, there are families who do not wish to be reminded of the past, and families who are reluctant to accept a weapon like this (for the legal reasons above, or just because they do not wish to own a sword)
It's very honorable that you would like to return it to the family from which it came, and I am 100% in alignment with that sentiment. Alas, the insurmountable difficulty of locating which Iino family it came from, together with the legal obstacles in importing weapons into Japan, make it very problematic.