My MIL is capable of doing it herself but it's been a couple of months. I have offered to help ("I don't need help") Suggested in the shower ("I don't take showers") Offered to trip to the salon ("I don't let anyone touch my hair") Took a photo and asked how she thought it looked. I there another way?
Only problem with this result, is even after father showers and washes his hair, his head is still itchy. So, now he thinks bugs are in his head.
The same applies when he didn't want to change his clothes. When he gets all itchy and scratches, I tell him it's because his clothes are stinky and dirty. He usually changes his clothes without fuss.
No one has died (that I know of) from unwashed hair. Maybe a little reverse psychology could work - as in just ignoring her hair. Suggest social outings, have people in, etc. and don't mention a thing about hair. Eventually, she may decide on her own to do something. Another thought is that if she has a good friend you can trust, tell her what you're doing. Then, maybe the friend will be able to suggest a join trip to a salon or something.
Sometimes, elders really hate being told - no matter how nicely or with how much love - what to do by their kids. They'll dig in their heels. But later, if we leave them to themselves, they'll "decide" to get a new hairstyle or just plain wash their hair or ask for help.
Good luck.
Carol