Hospital bed is uncomfortable.

My aunt has a great bed but I take her new sheets and pjs often. I have a hamper that I line with a large clear plastic bag. I do all her laundry. The extra fresh linens are in another larger hamper. Her NH would do all her laundry. I prefer to do it. You can learn a lot about someone’s life by doing their laundry.
I will say that the NH uses the knit sheets that look comfortable. You might also check the temp, noise and light of the room. So many factors to consider to get a good nights sleep. Some of which we have no control.

Get a memory foam and bring in your own sheets.
I've had lots of care clients who had hospital beds and they say the same thing.

You buy a piece of memory foam for a twin bed then trim it if you need to. That's what I always did.

If you bring in the sheets for your mother, you will have to be responsible for that laundry because the nursing home won't do it.

Buy a few sets of sheets that your mother likes. Put them in the bedside cabinet by your mother's bed. Then leave instructions that when they change the bed, you will come and get the linens.

I totally get where your mother is coming from. I sleep on 800 thread count or higher American cotton sheets and nothing else.

I bought a bed pad that cost a pretty penny. But it helped to keep the bed sores from coming and also kept my daddy from trying to get out of bed. I bought it at Bed Bath and whatever. I forget what it was called but it was great! Also, you can supply the sheets and write her name on them or sew a tag on them and be in charge of cleaning and changing them. I know it a lot to ask but those sheets are uncomfortable when you have to spend a lot of time on them.

If you can't change the mattress ask it they will allow an overlay - memory foam or perhaps a pressure relief one.

I’m sorry that your mom is having trouble getting comfortable.

What kind of bed is it? A traditional mattress or a rotating mattress that is used to prevent pressure sores?

Does she have any pressure sores that are causing her to be uncomfortable?