Family situations like this are seldom easy as you've probably discovered. If you were to say the wrong thing to your sister, for example, or approach her with the wrong attitude, you might end up making things worse instead of better.
That's why your first prayer should be for wisdom, asking God to help you know what to do. God knows what is best and He loves both you and your mother as well as your sister. God's promise is for you: "If any of you lacks wisdom, he should ask God, who gives generously to all without finding fault, and it will be given to him" (James 1:5).
Your sister's indifference to your mother's situation is wrong, and I hope you can encourage her to be more thoughtful, as God opens the door. The Bible says, "Do not withhold good from those who deserve it, when it is in your power to act" (Proverbs 3:27). It might be good for you and your sister to visit your mother together, then talk about how you can help her. Remember: Your goal is to help your mother, and not simply to get your sister to be more responsible.
At the same time, you may need to face the fact that your sister simply won't pick up her share of the load. In that case, ask God to help you overcome your resentments and do what you can to meet your mother's needs. Later, you'll be glad you did.