OP, I'm going to be perfectly honest with you.
I would have rung SS a long time ago, as well as offering every spare hour of support I could offer, I would have listened to my sister, and done whatever I could to help her to cope, rather than to prove I was right IYSWIM. I would be 'in it with her' rather than 'me being right and her being wrong' and I would be seriously worried that she had no friends IRL, and that her DP didn't help her himself.
I would have asked SS for some support for her, to help the DC stay with their mum, to support her in managing, not hoping she would be evicted.
It sounds to me OP like you have reached the end of your tether and the reason you are so frustrated with your sister is because she doesn't listen to you, doesn't continue where you have left off, or follow your example and it appears you feel your mum shouldn't help as much as she is doing.
I sense resentment towards your sister in each of your posts.
Has it ever occurred to you that the reason your sister may stay with her DP may not be because she is scared of being alone, but because she doesn't have anyone else who is not lecturing her on a regular basis about things she has no energy or motivation to tackle?
There are ways of helping people who are quite possibly very depressed, and showing them where they are going wrong and pointing out how smelly and dirty their house is, how unkempt her DC are and how they are not fed properly, or how she would be far better off without her DP, when she probably feels he knows her better than you or your mother does and maybe he loves her.
I haven't seen you say one positive thing about your sister, yet you seem to know an awful lot of what goes on in her house, even down to what they eat, how they sleep, who takes the DC to school, how often they wear clean socks even? 
You know so much, yet you come across that you don't even like your sister tbh.
SS can help, they will do what they feel is best for the DC or they should, and if they feel that there is a way of keeping the family together by offering support, then I think they would rather do that than separate the family.
If I was you, I would ring SS, then I would wait for my sister to ask for my help, and I would gladly offer it. I think your sister would benefit from some unconditional support to help her out of this dark deep hole she has found herself in, something which you don't come across as able to offer her.
If I was your sister, I would accept help from SS, and I'd cut the sister who resents me so much, loose.
I'm sorry to be so harsh, but I have not heard you say a single positive thing about your sister, her DC, her house, or her DP.
Just pick the phone up, and let someone who has a little more compassion and a little more understanding deal with a situation that you are too frustrated to deal with any more.
I'm sure I'll get flamed for this post now, but if I lived the way your sister apparently lives (and how do you know all these things, does your sister tell you all of these things?) I'd not welcome so much resentment and critisicm.
Seems to me you are more than half hoping your sister will fail and lose her home so you'll have the pleasure of saying 'I told you so.'