Can we make the nursing home clean our mother when she refuses to let them bathe her?

You bring an Rx from her doctor that calls for sedation and weekly bathing. A small dose of anxiolytic can make her calm enough to bathe.

I assume that someone in the state your mother is has dementia, so the idea of her having free will to refuse is questionable in my opinion. You don't ask an incompetent person to have a bath, you tell them, and if they need some calming meds on bath day you make sure they are prescribed. Generally when someone needs to be in a nursing home they require help with ADLs like dressing and toileting... you do mean a nursing home, not assisted living right? At the very least she can be sponged off and her hair attended to without a full shower or bath and her clothes changed every day as they attend to her other needs.

Sometimes, I think nursing homes just fall back on that as a convenient excuse for not having to deal with someone who proves to be difficult- although technically it is correct. A good NH will have developed methods for persuading a reluctant resident to allow a shower. My moms place had an employee who sole responsibility was to shower residents - how'd you like that for a job ? But I digress- this employee was sweet but firm and even though my mom hated showers - she rarely missed one.

Residents have the right to refuse showers/baths

We have found her dirty, she smells, her hair is greasy. This has happened several times and each time they tell us they can't make her bathe if she does not want to. Even when we tell them to bathe her!