She sounds like my mom who died recently of cancer but had Alzheimers for years before the cancer. In her own mind she is perfectly "normal" because people with Alz don't think or act the way we do. I was one of my mom's caregivers along with my dad. I know first-hand how difficult it is for you. I'll make a few suggestions that I'm sure will make your life easier, but they are suggestions only. If anything doesn't work for you or if you're not comfortable with anything, just ignore the suggestion. That's really important.
First, deal with the anger (or the bad days) by visiting a doctor and perhaps putting her on meds. My mom had begun threatening me but it wasn't normal for her to do that. Once we put her on anti-depressants, she was much happier and never again mean.
Next, deal with your own emotions. You will be in a much better frame of mind to deal with the daily problems of caring for an Alz patient. You have to go through the grieving process and come to terms with what is happening to your mom and also what you life becomes. It's not easy and there are some tough decisions to be made. If you don't have help, recruit family members or prepare to have someone who can pitch in at least on occasion.
Finally, try not to worry about her life passing her by. She will be okay in her own world. It's probably one of the hardest lessons to learn--her world is safe and calm and she wants to be there. It's not your job to bring her back to your world. I know, it's crazy in a sense, but if you can accept this one thing, you'll be okay.
In my experience trying to bring the person back to our world just doesn't work and you may miss wonderful opportunities. Mom would tell me stories about when she was little--her doll, her brothers, their games, etc. She spoke as though I was there alongside her during that time. It was just so strange to have these conversations, but I went along with her, asked her questions like what's the doll's name, what game are we playing and so on. (Although mostly I just let her ramble on.) I learned more about her life during these conversations than I ever thought I would have. And it's because I didn't remind her that her brother was dead or that I wasn't alive in the 1940's when she was a child.
Of course, by this time, she was in the more advanced stages, and I had stopped telling her who was dead and who was living. I never lied, I just avoided giving an answer that would be difficult for her. I had discovered that every time we answered a question truthfully and fully, she had to deal with the emotion and grief of the answer. She couldn't remember who had passed, so she would grieve for her father who died in 1972 or whomever she happened to be asking about. This happened every time she thought of someone who had already passed. I just couldn't put her through that. Eventually she quit asking if so and so was dead and started talking to him/her as though the person were alive.
In the final stages, she didn't even speak any more. And whether it was the cancer or the Alz that made her body slowly shut down doesn't matter. She lived a full life and I miss her terribly, but I would not exchange one moment of our time together. Even when she was difficult or in her own world, she was a wonderful person.
I hope this helps you. It surely helps me to remember her and to think that my experience may help someone else who is dealing with a loved one. Most people would say Alz is a horrible disease, that it robs the patient of life, but I came to terms with my mom's battle and decided that her life was worth my change in attitude. So, I changed my attitude and instead of watching her from the outside, I tried to join her where ever she happened to be. I'm not saying it was easy or will work for you, but it certainly did work for me and my mom.