Assisted living can’t make mom shower. Is there anything they can do?

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You need now to speak with Administration. There may be more than just this shower problem.
Your mother may be progressed too far now into dementia to remain in ALF. This is for more able people. She may require memory care or nursing home care at this point.

IF your mother is well enough to understand you taking her for a sit down talk with administration about cleanliness and hygiene requirements then she is able to understand your telling her that a failure to follow basic good hygiene habits will result in memory care or nursing home where she will be showered by caregivers.
If she can grasp what yu are telling her there's a chance here, but if she's incapable of understanding this, then she is likely needing a transfer to more safe care for her condition.
I am so sorry, and wish you the best.

Time for a meeting with the director to discuss the appropriateness of mom living there. Perhaps she now needs a higher level of care. She may also need medication for anxiety

Around 100AD, Roman villas in England had warm water showers. After the Romans left England, it was almost 2000 years until warm water showers became ‘normal’. Most of our own great-grandparents had a weekly bath. They did not stink, or have skin problems. Showers are NOT the only way to stay clean.

Why is she in assisted living and not memory care?

Despite what many people may think, We have to wash up somewhat regularly. If there are aides there that you know and trust ask them to give her a really good sponge bath. She can sit on the toilet and they can make sure it's warm in the room. Wash her from the top down and making sure that there are separate clean wash rags in washing front and back private spots. They can use a baby bath which will get her clean but not damage skin. This can be done about three times a week. Lotion can be applied on arms and legs and back. Dry shampoo can be used for hair, although when I was a CNA for home care people would let me wash their hair in this sink. She's afraid. So find someone really skilled and nice to help her through this. It doesn't have to be a shower she just needs to be clean.
Most CNAs who are skilled and loving will know how to give a good sponge bath.

M does not really need to have a shower. Old people don’t sweat much, and their care needs are really to clean around the genital area. M may be quite happy to wipe those with a cloth provided by a carer. That could lead on to more wipe-washes all over. This may be too much for AL, but it could be a quick half hour help from an outside carer.

I'm surprised they are allowing her to stay at the AL if she's refused to shower in a year. It sounds like she belongs in memory care now and not assisted living. Does she at least get washed up? It sounds like it's time for memory care.

I think your best bet here is to follow Alva's good advice in the comments. If your mother can still have a conversation have a sit down with her and the administration of the place and tell her that there are rules on hygiene that all the residents must abide by. If she refuses she will be kicked out and put into a nursing home or memory care facility. If she's beyond this kind of talk, she should be in memory care.

Do they ask or tell. When my Mom was asked "do you want..." she said no. But it they said "time for" or just took her, she was OK. People suffering from a Dementia can get like toddlers, you ask a question "do you" and the answer will be no.

I would give them permission to do what they need to to give her a shower. My daughter, RN, says you need to make them think they made the decision. "Mrs M, wouldn't it be nice to get all clean and smelly good and into fresh clean clothes?" Her residents usually say yes to that.

I agree with hiring a temporary CNA a few days a week to make sure your mom showers and washes her hair... I had patients that sometimes did not want to shower and I learned how to do a very very good sponge bath with them sitting on the toilet . They may do well together for quite a while until your mom needs more care.