How do I answer Mom when every time I visit she asks, "Am I staying here?"

Kudos to you for being there for your sister. I have learned a lot about dementia from Teepa Snow videos on YouTube. She gives excellent explanations about how it changes our LOs and how to better interact with them. She focuses on positive caregiving.

You can feel free to use "therapeutic fibs" to keep her in a calm state, she her dementia has robbed her of reason and logic (and memory) so that she can't sooth her own self anymore. You can tell her "I can't stay this time but will next time". She won't remember "next time". Resist explaining or reminding her of anything. Change the topics to other things that are pleasant, or take her to events & activities in the facility, or bring something she can do with you -- maybe a card game she remembers (and if she doesn't, do not correct her, let her set the rules). Maybe the facility will allow you to bring in one of your calm cats for her to pet. Sometimes dementia patients benefit from having a stuffed animal or baby doll to care for. My 100-yr old aunt in FL has a stuffed, multi-colored Llama that she calls her Baby.

You must change your assumptions about your sister's motives. She isn't the same person she was in your youth. You seem to be doing plenty by visiting her 2x per week. We visit my MIL once a week and I often feel guilty about that not being enough. Sometimes I wonder what the point is since she won't remember we were even there 1 minute after we leave.

Go live your life. Continue to visit your sister as you wish. Please do not feel guilty about anything -- you're doing as much as you can.

Tell mom, that this is her home, she is safe here.
Tell mon, you can't come home today. The doctor says you have to stay here where you are safe.
Tell mon, I am just visiting, I came by with a bit of dinner so we can visit for a while.
I do not think I would be a blunt in telling mom "yeah, you are going to die here"
Remind her that this is her home.
Trying to get staff to get her included, engaged in activities would be nice. But they can only do so much, if she does not want to join in they can not force her.
Are you 100 % sure that she does not interact with other residents? What she says might be different than what actually happens.

I am in your exact shoes. For years my folks needed to be in assisted living. They were told that by myself, my sister , their Doctors etc. My mom always said "NO, I'LL DIE FIRST" She had vascular dementia for a long time. I have devoted, I'd say, the past 5 years caring for them even though I live an hour away. Going there 3x a week. Brought them home cooked food, did their grocery shopping, took them for Dr appointments etc. I'm on disability and 62. My mom passed last month after going from hospital to SNF's and back. She was 86 I saw some dementia starting with my dad ( he's 90) I lived and took care of him for 3 months until I just couldn't anymore. They were married for 60 years. My sister (who lives close by but is rarely around) actually helped us find a nice place for dad. I got confused calls from him daily. "Who am I, Where am I "? And many even stranger phone calls with dillusions he's having. My dad who is extremely intelligent knows he's losing his memory. I took him often to see my mom wherever she was. We had to wait a couple weeks for when he wasn't so confused to tell him about my mom passing. I did actually stay with him for 4 days at the ALF sleeping on a blow up mattress when we moved him in. I was encouraged to leave and let them take over. So my sister has gone on about her life and visits once a week. I do nothing but cry all the time about plucking him from all he knows and placing him there, along with still grieving my mom's passing. ( main problem for both of them was living in a condo with 2 flights of stairs to get to their living space) Since dad has been in ALF, he always asks me to spend the night, which I can't. Wants to know when he's going home. They put him on Seraquil for his anxiety and being up most nights. Since then he started having a very hard time walking, taking baby steps, loses his balance. ( I've called the Dr. 3 times now without a return call). He's not at all interested in any of their activities. Can't be convinced because then he gets mad. He's never been that way my entire life with him. Now he's starting getting his meals brought to his room. ( which costs more) Always asks if I'm going to eat with him. I have on several occasions but try to leave before they start coming around because if I say I'm not going, he won't either. ( I'm supposed to make a reservation 24 hours in advance if I am to eat with him) My home isn't large enough for the both of us. My sister wouldn't think of it. After all the good he has done in his life, he's placed in a home. Even though I've decorated it with his favorite things. It's still not home. And he's lonely. I too see him getting worse since he moved there) He only wants family around. Doesn't want to meet anyone, form any friendships or listen to their " blah, blah, blah ." He just isolates himself and most of our visits are me watching him snooze in his chair.
Should have made this a separate post. I know exactly how you feel, believe me. I know when I visit, I could very well get a call the next day, or that night even,, wondering why I haven't been around. It's myself, my sister and my brother that lives in another State and is homeless. (His choice). So really, it's just me. To not go crazy anymore I have had to change how I react to what he says. Before I would go to him immediately. ( Well within the hour) Now I don't answer everytime then listen to the voice-mail and then call the facility to check on him. That's what I was advised to do. When he asks me to spend the night, I just change the subject as many times as I have to. He'll also say, " so what? I stay here"? It's been 4 months for him. I suffer from such guilt that this is where he's at. I don't remember what my life was before all this caretaking. This should have been a post on its own but I just wanted to let you know that you are not alone. I get those same questions and struggle with them too. Our stories, so simular that I just had to let you know.