Trying to get father out of nursing home. I'm the oldest child.

I am going to assume here that you have been on Forum a while, or have experience that allows you to understand what 24/7 care is. Because like all the others here, as bad as you imagine it to be it is 100 times worse. It will rule your life during his lifetime.
That said, and assuming you know the sacrifice, we are down now to your FIL. Apparently he is still married to his wife? She then would be the one who put him in care? She would be the one to contact then and offer your home and your life dedicated to the care of her husband. It is up to her. Again, assuming that your husband doesn't have POA.
May I ask how long your FIL has been in care? May I ask what has led you to make this decision for your own and his life at this point? Just curious.
So assuming there is no divorce, your FIL's wife is the Lioness at the Gate.

NO!!! Don’t do it!!! Don’t bring him in your home!! It won’t be a fun and rosy time!! You will have to care for him 24/7 with NO time for your husband!!

If FIL is in a NH then it has been found he needs 24/7 care. You really need to understand what 24/7 care is. Someone needs to be with him at all times. If you work, you will need to hire aides. If not an agency (expensive), and you private hire, you are responsible for payroll taxes and need to pay a reasonable hourly rate. Then you have the aide not showing up on time or at all. We had one member lose their job because the parent couldn't be left alone and the aide was always late. Neither the agency or the aide were sympathetic. And companies are not nor need to be sympathetic to your problem.

And who is going to do most of the care. We have had members post that a DH has brought Mom or Dad home to live and felt the wife should do all the caregiving. Do you want to be responsible for bathing, toileting and dressing ur FIL. We just had a man post that his wife brought home her Father and expects the husband and the kids to care for him.

Then its if your house is handicapped accessible. Do you have a walk in shower with bars on the wall. Is the toilet easy to get on and off.

If husband does not have POA then his hands are tied. Guardianship would be the next thing and that is very expensive.

As suggested, you may want to talk to the Director of Nursing to see what is involved in FILs care. Seems like he has Dementia. Taken him from a place he is familiar with and moving him with "strangers" because if he doesn't know you, that is what you are, it not good. And Dementia is unpredictable. You never know what they will do. So before you make this decision get the facts and realize you will be giving up ur life. If u have or plan on children, they should come first. A person suffering from Dementia is just a big child.

There are a lot of things to consider here. Are u really prepared to take this on ? While it sounds noble there is a reason he is in a facility. Have u asked facility if u can check him out for a temporary visit so that u can access the level of care that he needs ? I took care of my mother and mother-in-law for 20 years, worked full time and raised a family and it mentally and physically exhausting.

You would have to file for guardianship. Is he a ward of the state now?

Don’t take him to your home. You have no idea what you’d all be taking on.