What is a fair amount to pay for personal care and rental agreements for their parent who will live with a daughter or son?

Not that many clothes, she wouldn't. Your profile lists incontinence among your mother's care issues. Her share in the weekly laundry workload will add up to a lot more than "just one more person in a household."

But I agree, it is a complex issue, and not one it's impossible to quantify precisely (fortunately you're not expected to, you're just required to give it your best, realistic layman's guess).

There are the living costs - your mother's share of utilities, groceries, disposables, repairs, maintenance that are part of the routine household budget.

Then there are the direct costs of goods and services that will be solely and exclusively required to support your mother. This might include continence care supplies, toiletries, foods and drinks that she prefers, small ticket items such as chair pads, sippy cups, bedlinen, hand rail installation, lighting, a turntable cushion to help her get into the car.

And then there's the care - the time, the unsocial hours, the 24-hour responsibility. The work, in short. How do you think your sister's going to bill that? If your mother doesn't want her breakfast but does fancy scrambled egg in mid-afternoon, does that count as two catering charges or only one? When the one hour allowed for showering turns into three. When it dawns on your sister that there isn't a figure that compensates for broken nights night after night after night, or never sitting down to eat a meal uninterrupted.

I suggest you add respite breaks and paid professional hours into that paper trail. There isn't a fair amount that will cover the cost of care, because money soon stops being the critical issue.

Your estimate will be based on the present requirements. Your mother’s requirements are only going to get more, so what is reasonable now can change in the future – quite quickly as well as in the long term. I remember a poster who had agreed an amount they thought was generous for the family ‘carer’, and then were angry when the ‘carer’ quit as things got worse. Poster thought the agreed rate should cover all eventualities.

Perhaps make a temporary arrangement, with provision for reconsideration when there is more experience, and regular reviews after that. That’s also a warning against spending too much money on modifications to the house, before you have more idea about how it will go. Lots of things to think about!

You need to start with rent, utilities, food and take annual inflation into account, you can base it on national average from previous year, those are expenses your Mom would incur living independently.
Saying your sister is doing laundry anyway seems a little presumptuous, she will have to do much more and pay more for hot water, electricity.
With average disease lasting 10 years, and most are progressive everything will become more complicated, more appointments, more meds admin, dietary considerations.
And the need would be for respite care, your sister in the long run cannot do it alone, her compensation should reflect and take into consideration potential time spent helping Mom at night, those positions based on getting up 1-2 a night are generally about $100 per night.
So her pay should have increases as well, saying $24 per hour today, it is somewhat low, her salary should have exponential increases, either based on increases in responsibility or you can base on caregivers salary in your specific area or national increases, those are widely publicized.
My calculations shows about $10,000 a month is not extremely high for amount time and range of responsibilities, including but not limited to cooking, cleaning, medical appointments, personal hygiene.
Alternatively, you could make 5 years plan A if it is not feasible to provide care at home, considering other solutions.

Sister is taking on a Herculean job. It's 24/7. There will be crises in the night, and how do you pay her for lost sleep? How do you pay her for special laundry issues, like poop stains on the underwear? The urine smell that may pervade her house? How about withholding for your sister's tax purposes, how will you account for that? I paid dad's live-in caregiver $275 per 24-hour day 10 years ago. Then we had relief caregivers to help with the lifting and filling in for the 24/7 live-in. That was about $20 or $25 an hour. I was there too when needed. Who will help your sister with such things? Who will do the housework? There's quite a lot more mopping and swabbing with a sick person in the house. Who will clean the carpets? Who will stay with the patient while sister goes out to buy food and medical supplies? How much do you pay extra to sister when she's spent all morning on the phone dealing with mom's issues for the doctor, the pharmacy, the credit card that has been denied, and all of that while mom is calling from the other room for a drink of water or lunch? You'll need to address all of this, and God help Sis. I mean it.

Mobility and incontinence problems?
Hire an outside caregiver to come to sister's home on a regular basis.
Maybe 4 hrs. 3 days a week.

1) Separate the expenses from the caregiving. Under expenses: Divide ALL the household expenses (rent/mortgage, utilities, maintenance, even taxes) by the number of people in the home. Or if children live there, divide the expenses by the number of used bedrooms. Adjust that expense to what Mom would be reasonably able to afford on her own, elsewhere. For example, what has she been paying for rent?

If you say, Oh we could never charge our Mother rent-you are just not getting the paper trail concept. But you will. For record keeping, you may want to keep this figure high. The caregiver who does not want to enrich oneself can choose to spend the money on special things not otherwise provided for Mom-a night's stay in a hotel/short vacation maybe. It should not cost your sister to take Mom in. Controlling these funds, your sister will not be forced to run daily caregiving expenses through the POA.

2) Caregiving: The amount will change. But caregiving is a strange thing. The caregiver may be plagued by always checking on the loved one, 24/7, always thinking about her care needs, and never really being able to relax or have time off. Even though you may not pay for this, the caregiver will always be 'on duty'. And, the caregiver will not be free to earn a living with contributions to her own Social Security/retirement.

Generously pick the hours of official on duty, then the hourly rate.

Provide "respite" care funds so sister can take off.

Depending on the needs of Mom, consider that a nursing home can take all of a person's income, less a personal allowance. Consider that AL can be more expensive than your Mom living at your sister's home. $2800-$6000 or more.

I have adjusted my advice to you accordingly because of these words:
"Just another person in the household".
"Sister would be doing the laundry a n y w a y ".

If your sister were asking this question, I would advise her not to take this on if siblings are going to be POA. She should be POA.

Maybe as an exercise, you could calculate what you would charge Mom to live with you, and if you had to give up your job to care for her. How much would that be?

Apologies if I did approach this "as if" biased in favor of your sister's income.
Maybe changing the words "fair amount" to a "generous amount" (Mom wants to be generous towards her caregiver ) would help you decide.

Verify County eligibility.. Try Meals on Wheels.

Pay your sister fairly. Yes, she's washing but go by how many extra loads she will have to do for mother. Take into consideration her loss of privacy, etc. How picky an eater mother is. Does she have to eat on a certain schedule. There's way more to it than just one more person.