How do I handle caring for a difficult parent?

Our mother is 76, has been widowed for 4 years. Flatly refuses to do any housework or even try. She is not disabled, has mild cognitive impairment including diabetes.
Our mothers aptitude towards my sister who is her main carer and anyone else stinks and is very hurtful and selfish. because she has this opinion that life is lived her way or the high way. All our lives she has refused point blank to do any housework at all. When we were kids, Dad drew up a rota of household chores which I and my 3 siblings had to do every day. We did everything!
This is not new behaviour for mother, she is essentially lazy and always has been. Dad and Mum used to have terrible arguments about her laziness.
their relationship was very destructive and us kids tended to ping pong between the two of them. She's a master at manipulation and deceit and yet everyone thinks she "Presents Well" and says how lovely she is (outside the home).

Since Dad died, it's like she has decided to sit on her backside all day and let everyone around her do everything for her. All she does is eat and watch TV.

We have tried talking nicely to her about it, we have tried to encourage her to help when we do her housework but she flatly refuses and her stock reply always is " I never asked you to" Trouble is if we don't her house quickly becomes a health hazard. She will not wash up, and will happily leave dirty crockery in the sink. We bought her a dish washer but she refuses point blank to stack any dishes in it. Once my sister and I refused to do any house work what so ever, Mother didn't care. she used every plate, cup, knife fork and spoon and when she ran out she started washing the dirty plates under the cold tap. The house looked like an absolutel tip, stank and in the end, we relented because we were worried she'd become ill. Her eating habits are a different story and one that we have finally given up on. She eats rubbish all day long and says she cant remember when we try to stop her from over eating. We gave up trying to regulate what she eats and just let her get on with it.
Her aptitude towards us has and always is, that Children should be seen and not heard, Respect their elders and that the reason she had four kids was for a steady supply of servants.
My poor sister is at her wits end. Sadly she lost her temper with mum the other day. Read mother the riot act only to be told to get out. Which she did, and rang me in tears. Mother tried to apologise by grabbing my sister and demanded that she let her hug her. My sister said it was all she could do not to push her away.

I was up at mums yesterday, she was at the day centre, when I arrived the house stank of dirty old women. Mothers breakfast things were left on the side, tea stains were all over the work top and there was washing in the washing machine which my sister had put in the machine two days ago, asking mother to hang it out, she didn't and it smelt. The dishwasher was full of dirty and clean crockery mixed together.
Both of us are feeling abused, disrespected and completely fed up with her behaviour. Nothing seems to get through to her and what's made matters worse, we know that mother has not washed for 6 days now. We have tried everything from bribery to just being nice to her. When she came back from the Day Centre, she was covered in talcum powder, her flies were undone and the top she wore wasn't even ironed! I tried to have a chat with mum to find out why she feels its ok to live like she is and to let me and my sister be her servants. How stupid was I, as I got, I never ask you to and if your going to have a go at me you can get out too. so I did. She says that she likes her life and does not see what or why we are so anal about her personal hygiene or dirty house. Granted no one will die of a dirty house, but when it starts to affect her health and wellbeing and she's not washing, what are we meant to do. The not washing bit is her having a paddy because my sister called her a smelly old cow! We're in our fifties!! is that anyway to speak to your mother?

Yep! Small wonder we don't feel like spending time with her or taking her out. When we do, she makes it obvious if she does not like where you are taking her, by deliberately walking slowly and bumping into things, but if its something she wants to do, she'll stand up all day without a complaint!
She's been tested for depression and assessed by the social services. It was our choice to care for her at home, but now we wish we didn't take on this burden,
Small wonder that both of us have said we wished she'd hurry up and shuffle off her mortal.
No sympathy please, we're past that. but any practical tips would be a God send. Thank You.