Six months ago we moved my Mom from a NH in with us, now she doesn't want to stay. Should we take her back?

When we moved her here she knew who we were. Now that her dementia has progressed to moderately severe she doesn't remember us anymore and doesn't want to live with strangers. There are no other family members that can take care of her, which means she would have to go back to the nursing home. After getting into an argument today over who I was and trying to defend myself saying I was her daughter, she thought I was lying and tried to leave. Even though she didn't get very far up the sidewalk (using her walker with me watching to make sure she was okay) and then me having to walk her back after she almost fell, I feel like this is a no-win situation anymore. Of course, every evening she wants to go home (home being her hometown in another state where she grew up with her mom and dad who are long deceased and there is nobody there anymore). For a while I was able to deflect by telling her it's too late and we'll talk about it tomorrow. Now all of a sudden she has started wanting to go home in the afternoon. So I try to ask her where her home is and how's she's going to get there. Of course, she doesn't know. Now she's starting to tell me she doesn't want to play any more games. Then today she started talking about her daughter, Lori, and I told her "I am Lori". She said, "No, I mean my daughter, Lori" to which I said, "I am your daughter, Lori." I know I probably shouldn't have argued with her about who I am, but I was just getting frustrated over the fact that she wanted to live with us 6 months ago and now she doesn't remember who I am and doesn't want to stay. Then she just got mad, grabbed two jackets and put them on one over the other, and headed out the door with her front-wheeled walker. I let her go because I knew that she wouldn't get very far without either getting too worn out to walk any further or possibly falling. And I thought, if she ends up falling then maybe she could go to a nursing home for rehab for a while! Otherwise, it's just going to drain all of what little money she has left which is supposed to go to my brother and sister who are both disabled. Anyway, I decided to watch her from the front porch just to make sure that she didn't get into any trouble, and sure enough she saw that the sidewalk was ending up ahead, so she tried to take her walker from the sidewalk down a slope to the curb so she could walk in the street. As soon as I saw that she was going to fall I ran up and helped her back onto the sidewalk and walked her the rest of the way home. Again, by this time she was getting worn out and started getting shaky and having difficulty walking. We made it back to the front porch and I let her rest for a while before we went back inside. I thought this would all be a distant memory by tonight, but then when I took her in to help her into bed she said that she wanted to leave because she "didn't get along with the other gal too well that was here earlier." I told her, "That was me!"

So my question is, do you think I should take her back to the nursing home or continue to try and redirect her to stay here? Has anyone else dealt with a loved one that doesn't remember who they are anymore and therefore does not want to stay with you? I'm finding that I am getting very frustrated over the whole ordeal and would appreciate any suggestions you might have.