You are dealing with one of the hardest situations to cope with and it takes both courage and patience. My mother suffers from Alzheimer's and I am so very familiar with the conversations that you describe. Repeating the same question and answer over and over again. It is no accident that caregivers like yourself are at high risk for alcohol or substance abuse. It is hard physically, mentally and emotionally, profoundly so when it is a parent that you are caring for. Your mother is very blessed to have you there to care for her. Many are placed in facilities that do not have the resources or the degree of love that a family member would have for them.
Recently both of my step-children had dental surgery and on the ride home I had the same conversations with them, over and over, too, because of the medication that they were given. My stepson kept saying, "why is my mouth numb?", my step daughter kept saying, "why do I have this gauze in my mouth?". It struck me that these were exactly the kinds of conversations that I have with my mother, too. It helped to remind me that it is just how the brain works. I have to remind myself that for her, each time the question is new. If I'm angry, she doesn't understand and it only adds to her stress, causing her to remember less.
Sometimes I find different ways to answer her, all of them honest, and try to treat it like a game. Other times I've learned to distract her. I have found that, like with 4 yr olds or puppies, it can help to have a few attention-getting items handy. A rosary, a book with pictures, sometimes just whatever I have nearby. Photographs work really well, have some old photos handy, especially older photos of her when she was young. Chocolate works really well for us since Alzheimer's patients have a sweet tooth. I realize this sounds patronizing, but if I can find a way to sort transition to something else and not get angry, it keeps her from becoming stress, too. For me, it's as though my mother is four years old again and she needs that guidance and coaxing that one would give to a child. It helps her to feel safe and happy. Somehow I find it comforting that she is returning to God like a child. There is a sweetness in that.
The anger is perfectly normal and mirrors my frustration at the whole situation. I, too, had family to deal with that complicated everything both practically and emotionally. Being there all day every day feel so overwhelming and leave one feeling trapped and helpless.
There is help. In our area there are drop in day cares for elderly and there are organizations that provide volunteers to give caregivers like yourself a day off.
There are also support groups, particular for those dealing with dementia, or for family members of those who are mentally ill (dementia). Dementia is dementia and Alzheimers just happens to be the kind that gets most attention, but there around many kinds of dementia. It doesn't matter if she's specifically diagnosed with Alzheimer's, the support is there for you. This might be a good starting place: alz.org/apps/findus.asp or check see if there is a National Alliance for the Mentally Ill (NAMI) near you. Some NAMI sites have online chat rooms for support.
These are perfect places to vent, so that you do not carry all the pent up frustration. Some programs give caregivers a "day off". If you can find a program to give yourself a rest, I pray that you can find a way to do that. Can you have someone sit long enough for you to go out to the movies? have coffee somewhere? You absolutely must take care of yourself if you expect to be of help to anyone else. Please know that you are not alone in feeling the way that you do and that the feelings are normal. Be compassionate with yourself. You, too, are a child of God. I pray that you will be able to find some support.
I often turn to Mary and Joseph, the eternal examples of patience and steadfast love. Grant you, I don't emulate them very well, but it gives me a goal to aim for. I hope that some of this helps. God bless you for all that you do.
May God bless you with peace and comfort during this difficult time and pour blessings down upon you for being there to care for your mother. May He give you the Grace to be patient and loving, even on the more difficult days and help you to see that there is still a whole, beautiful world around you, with sunshine and laughter and Hope. May God continue to keep your mother content and bless your sister, too. May God guide all of your family in their actions and their emotions and may He use all these things to bring each of you closer to Him and may you feel His great love for you.
In Jesus' name, Amen.