What kind of help is it you are hoping your sister can offer you? If you let us know where it is you are strugling then perhaps we can offer some solutions to help you get through this really tough time.
Families should be there to help you yes, but the reality nowadays is that they aren't. And many people will defend her right to her own life and that you shouldn't expect her to help you rather than consider the fact you are desparate for help and have no where else to turn too. Many people are selfish and put themselves first in our modern society. They expect the state to give you the help and support you need rather than put themselves out.
I used to always be there when my sisters needed help or wanted anything. I would send them food parcels when they were poor students. I hired a vehicle and moved my sister and her then bf to the other side of the country when they couldn't afford to move and lied (by omission) to my parents when she insisted I didn't tell them certain things when she fell out with them.
I have changed my work days so I could look after my other sister's dc on many occassions. Driven an hour each way with my ds to look after her dc on bank holidays when she had to work etc. Taken her dc out so she can have a break and bent over backwards to help her, despite her being married with a husband who is supportive, and I being a single parent.
On one occasion my other sister phoned me late at night in tears asking me to come and collect her after rowing with her then DP. I was 8 months pregnant and there were gale force winds but I still got in the car and drove half an hour each way to get her. I thought her DP must have hit her or something she was so hysterical, but no, she was just upset and emotional and when he suggested they needed some space she called me to come get her. I didn't say anything I simply drove her home.
The last time I had my sister's dc I asked if she could bring them to me (not unreasonable I thought seeing as I was doing her the favour and couldn't afford the petrol nd had gone and collected them on numerous occassions before). She was hugely put out and in the end I agreed to take them back to her's if she would drop them off..! Not once has she offered me petrol money.
And there was the time a few years ago I told one friend I was feeling suicidal and really needed support and help. She didn't contact me for weeks, and it was in fact me that contacted her first - again... Although she did come good for me another time.
All my help and support to family and friends over the years was mostly unreciprocated. My expectation was that my sisters would help me too when I needed it, but the reality was that they didn't. The resentment has built to a point I no longer talk to one sister and am on strained speaking terms with the other. My sister feels I am selfish for wanting to be a part of her life and asking for help rather than waiting till it is offered (well I've been waiting a damn long time for it to be offered!).
When I complained that she withdrew an invitation to my ds for an outing for her ds's birthday she told me I was selfish for wanting to be there and that she doesn't regard me as family - her DP and her kids are her family she says. Even though she still had grandma along for the day. She said it was her ds's decision at the age of 5... 
although her ds said 'mummy said...' when I told them how disapointed ds was at having his invitation cancelled.
Letting go has lifted a huge weight from me. That constantly being let down and being made to feel like an imposition for even asking for help has gone. I no longer expect any help or support from anyone because the reality is I won't get it so stop expecting it.
You're right that it shouldn't be this way, but I'm afraid that you will have to find a solution without hoping for your sister to help you.
From what you say I think she has backed off and isn't phoning you because she knows you will ask for help and she is unwilling (for whatever reason) to give it to you.
Perhaps you should break the deadlock and call her to see how she is. If you think it is worth the effort to maintain a relationship. But don't ask her for help and don't expect any to be offered. If she asks then say you are struggling and really need a break, but don't expect her to offer to do it for you.
Do ask your surestart centre what help and support they can give you. And look for suport groups and help in your area. There are charities that will come and visit to simply give you someone to talk too or help you with some chores if that is what you need. They vary from area to area though.
Unfortuntaley most people have to reply on this kind of help and support nowadays rather than that of family :(