What is your question?
When I was in Japan, I was very concerned about the fact that none of the public bathrooms I went in had soap....and I don't mean the dispenser was empty, I mean they had no soap or towels with which to clean your hands. Even when going to peoples homes, there usually wasn't soap available in the bathrooms. (I carried hand sanitizer and wipes). I still wouldn't be so quick to say that it's a cultural thing, since I didn't go into every Japanese home, and I also know many American homes or public restrooms where I didn't see soap as well. I was told that many Japanese carry soap and hand clothes with them.
If you could explain to her about transference of fecal matter, and how germs can go from the hand to the doorknob, counter, faucet and then get on someone else's hands and make them sick, or even make her sick just by being on her hands. Perhaps you can put antibacterial wipes in the bathroom and she can at least wipe her hands when she's done. You can use antibacterial wipes to wipe the counter and faucets before using as well. I noticed a few Japanese that still carry handkerchiefs, so maybe she would cover her mouth with these if she had some?
Perhaps a night light that would stay on in the bathroom whenever it got dark, something bright enough to see by would help? Regardless of culture, it's very hard to change older people's habits. However she has to realize that it's not just her living alone and she has to show respect to you as well by not making you sick. When someone is sick, washing their hands can be the difference between just that person being sick, and everyone is the house being sick.
I hope this helps. I welcome your thoughts, let me know if you want to talk more.