Be direct where you can, do nothing where you can't
Answers on IPS require either cited authorities or personal experience. I'll answer mostly from experience.
Sometimes, by the time relationships get out of control to the point where they need a letter, the letter won't do much good anyway. Sometimes it can help—and we must do all we can do to help—, but even then, if a letter is in order, you must accept that you won't get all the victories of friendship that you'd hoped for.
Look up incorrigible and Some People Can't Love.
But, no one is hopeless who is still breathing.
1. Respond Rightly: Don't try to change your sister! Don't try to convince her of anything! Just have the right reaction of decency as an encouraging and inspiring demonstration for both her and your mother.
Another factor is The Savior Complex, where we try to "help others" when maybe we shouldn't. Sometimes, "helping others" is a kind of mental illness, but I don't have the licences to diagnose that.
Maybe you and your mother both have been trying to help your sister too much. Maybe you need to take some respite from that and focus on communicating to your mother that she—your mother—can love your sister more by not trying so hard. Love knows when to let go.
2. Just Love Both: Focus your energies on loving both your mother and your sister, just as they are. It might be your turn to be the adult in the family.
With those two in mind, you might be ready to write your sister a letter. Why not write a letter to your mother while you're at it?
I've had to write letters to family members now and again. I know a teenager from a divorced family who wrote to his (difficult, possibly certifiable, often dramatic-neurotic) mother,
"Mom, I just want a good relationships with you. That's what I want. I don't want to argue or be right, I just want a relationship."
That changed everything. She was still difficult much of the time, but they were able to do things together and get home in one piece, which they couldn't do before.
The teen added a few more words, but they were all original. No one gave the teen inspiration for it, the message was as original as it was from the heart.
3. Write for Relationship: Speak the truth, call out some things that need to change, but mainly write a letter asking for a workable relationship.
You can speak the truth on some things, but make "relationship" your letter's priority.
Maybe your sister's life really is busy and difficult. Maybe she is just occupied working, paying bills (job hours), or zoning out trying to process the flood of stupidity in the world. Maybe you can give your sister your mother's permission to go away for a while and deal with her stuff. Whitney Houston sang, "If I should stay, I would only be in your way..." — I Will Always Love You. Don't go away forever, just a little while. It's not "being together" that must be forever, it's "love", which isn't hurt by the need to lend time and space.
You haven't helped mom around the house much, yet she has scaled mountains for you. No one expects a thanks, but please just silently notice her love for you. And, know that I love you and so does mom. We'd love it if you'd drop in once in a while. If you can't, that really is okay also. We just miss you and want you to know.
You should be able to write something like that without violating your mother's trust. If you can't, then there are deeper issues of honesty within family—to tear down the stonewalling—that you and your mother can discuss while you care for her. But, maybe that's not a problem, hopefully not.
Sometimes, it's the heart that makes the body sick anyway. Whatever is going on, help everyone change themselves—no trying to change the others. Do that by not trying to change others, only improve yourself. Pull out your A-game with this. Don't whine to yourself that you get no help. Shine love. Share your "my-role, love-others" thoughts with your mother and thus inspire her swift recovery, along with the whole family's.