How do we confirm that the Memory Care place that we are entering will keep her when the private money runs out?

You are best to speak with the Facility itself. They all vary. They will discuss all of this with you by a phone call. It is very valid to worry about, and you should have all these questions answered for you.

This is up to the memory care facility and they well MAY NOT let the person stay when the money is gone. It was made CRYSTAL CLEAR to my bro when he went into his assisted living that--not put this bluntly but definitely made clear--when the money is gone so is he. This in fact was in the very long contract with them that deliniated EVERYTHING and which had to be signed page by page. They told what the levels of care were, how much they cost, when you would move to another level of care, what the memory care was, what it cost, how much you could expect in raises by the year (3% to 5% barring the unforeseen, and covid is that). Everything was in black and white. This is what you need.
A facility such as this is not regulated by the laws that nursing homes are regulated by. It is a business. When your money goes, you go. And there may be SOME that allow you to stay, but anymore, it's rare as hen's teeth.
The ones to ask are in administration at your facility. If you do not have this in black and white, a signed contract, then you have nothing.

My LO’s contract with her AL contains that language. Read every word of your contract before (her POA?) signs it.

Not sure if this is formulated within state or federal laws, but I feel fairly safe with the document in hand. May not apply in Minnesota.