There may be no “easy” way to deal with your family situation. Your sister seems loyal to your mother. Even if being loyal to your mother means that her quality of life is severely compromised there may be nothing that you can do to change this. As you said your sister does everything your mother says. She’s under your mother’s control. Your sister cannot make a decision without consulting her. You cannot speak to your sister without your mother being present. Maybe your sister will eventually come to realize the damage your mother may be causing her but she may have to come to this understanding on her own. This process may take days, months or years. Be prepared for the possibility that your sister never comes to this realization. You can be there for your sister in case she ever needs support or advice but for now you might have to accept the relationship between your mother and your sister as it stands.
I know that you wish your family’s situation would improve. It’s unclear whether or not they can change. Your mother seems to have many mental health issues. This may make it impossible for the family to be fully functional and cohesive. Your father excuses her behavior probably because it’s easier for him to believe that “is the way she is.” There may be many other family dynamics at play as well that were not described in your letter.
Another major concern is that you and your husband own a business with your mother. As you said, your “livelihood” is dependent on her. This is an untenable position to be in. In the near future it would be wise to explore how you can cut ties with your mother and this business. I do not mean that you should sell your business but you need to find a way to regain control over your financial life. You do not want to be inextricably reliant on your unstable mother for your income. You need to protect yourself, your husband and your two children from your mother’s instability. I would suggest contacting a lawyer regarding this business matter.
As I mentioned in the previous paragraph you may not be able to change your family situation. Your sister may always be reliant on your mother. Your father might always make excuses for your mother. Your mother may never change her behavior towards you or other family members. Unfortunately you may never have the type of family you desire. You have to deal with the “hand you’re dealt.” In order words, you have to deal with the reality of your family situation.
You may not be able to change your family but you can learn new ways to deal with them. It’s important to learn how to protect yourself, especially from your mother. For instance you cannot change your mother but you can change the way you react to her. It’s important for you to learn not to give into her, how to handle her when she makes unreasonable demands, how to place boundaries on what you will and will not do for her, what you will tolerate from her, and so forth. If you need coaching on how to deal with your mother then I would suggest therapy. A good therapist can advise you about more constructive ways to handle her and the rest of your family. Your husband might benefit from counseling as well.
It’s not going to be easy to change how you deal with your family but you can if you try. I should also warn you that if you change the way you react to your mother she’s probably not going to like it. It seems like she has always had control over much of what you do and how you think. If you begin to change the way the two of you interact she might feel like it’s a threat to her control over you. It’s a complicated situation and that’s why you should consider seeing a therapist about this matter. It’s time that you take control of your own life. Therapy can help with this.