My clients don't set my fees, I do. My fees are fair and yet, periodically, someone will always so up, wanting a discount because they have more than one child, because they can't afford it, because someone up the road is cheaper, because there's a "Y" in the day.
I not negotiate fees, hours or terms. This is the service I offer, these are the hours I will provide the service and this is the cost. If they not like any part of that, they quite welcome to go elsewhere.
This woman would understand that something Sears sells will cost more than Walmart prices because of quality of product. She would not walk into Sears and demand Walmart price. She would not walk into Walmart and demand Dollarama price.
My experience is those want cut fees, if they come to you, will question many other things. I suggest your contact is water tight. Be clear if all day care is available for snow day, March break, other closures, what hours it is and what all day fee is. Also if you not available all day if school closed, make sure contract very clear on that.
If she want your service, she pay your fees. If she feel too expensive, that is entirely her prerogative but you not obligated to cut your income to fit her budget.