As I see it, there are two options open to you:-
1. Asking for a divorce, citing his adultery with your sister as the cause.
2. Learn to accept that your husband can't keep his dick in his pants, and your sister can't keep her legs closed. And ignore what they are getting up to.
You can try insisting that he stops having this affair with your sister, but that hasn't worked so far as it!!! You have fought with him and her about this, and it isn't preventing them from having sex with each other.
The disharmony that this is causing within your household is affecting your children. So will divorce. But, if you are so unhappy with the way things are now, then even if he does promise to stop, you will always be suspicious of him, the moment he steps foot outside the door, or whenever your sister is in the neighbourhood, or just when you are sitting quietly, thinking about nothing in particular.
When people start having affairs, the last thing they think about is the people that will get hurt by it. And once you have lost your trust in your husband, you may forgive him these misdemeanours, but you will never forget. Every time he does something questionable, every time he does something that annoys you, every time you want to score points against him in an argument, you will remember how he has betrayed your trust. Furthermore, every time you see your sister, you will get those painful feelings in your chest, and probably, your fingers will itch to scratch her eyes out!!! Your children will notice this, and it will upset them, knowing how their mum feels about their dad's betrayal of her with their aunty. Divorce at least gives you a clean break, it lets you get on with your life, it gives you a sense that you have rejected the bad parts of your life.
People are always afraid of the social stigma attached to divorce, but this is really not worth considering. You don't have any parents who would insist on you maintaining their honour, and your own honour has already been ruined by your husband's betrayal with your sister. I can't imagine that anyone would expect you to remain in a relationship so destroyed by disloyalty.
The other thing to consider is this: all the time that you are fighting with your husband, your children are affected by it. A clean break from him would at least put an end to the fighting, and the children could then settle down with one parent who gives them undivided attention and love. If they wish to, they can still visit their father, but if they know what the situation is at home, I don't doubt that they are disgusted with him and their auntie.
As I said, the second option is to accept that your husband is unfaithful, and with your sister. You can continue living in the same house, but effectively, your marriage in the real sense of the word is already over. He can tell you all kinds of things about how much she loves you, And he only has sex with her because he lusts for her, but if he really loved you, would he have betrayed you with your own sister?
Men and women have affairs, it's a simple fact of life. The reasons for it are usually because it is exciting, or the sex at home is not either frequent enough, or has become routine and boring. Some people have affairs simply because they have the opportunity to do so. Finally, you have to accept that love doesn't always last forever, and some marriages were never made for love in the first place. In your case, whichever person seduced the other, the result is that you have become the third party, while they are busy having sex with each other. I don't know what it has done to your own sex life, but I would imagine that every time he even comes near you, all you can think of is of him getting out of your sister's bed!!! If you can learn to live with this, and let them carry on their affair, then your marriage could at least remain intact as far as certificates are concerned. It still wouldn't make your children happy, because they would still know what's going on.
I don't see any point to involving your older brother. If he lives in America, he has probably got a life of his own, and he probably has a very American way of looking at life. He would probably be telling you to get a divorce, and sue your husband for as much of the communal property and assets as possible, as that is the American way. Personally, if he recommended that, I would agree with him.You are the victim here, and I can't imagine any divorce court doing anything but finding in your favour.
You have been betrayed in the worst possible way, by your own sister. I suggest you tell the pair of them to get on with their lives together, and be gone from your house. If he really loved you, he wouldn't have done this to you. It would be better for all concerned if he went and lived with your sister, and left you to get on with your life.
As for your kids, whatever course of action you take, they will face upheaval, and they will have to cope with it as best as they can. Believe it or not, kids are pretty resilient, and they can get through their parents divorcing and splitting up, without it destroying their own lives. Don't you think your son and daughter would rather see an end to the fighting, and you at least coming out of the divorce, knowing that you have done the right thing?
I invariably try to answer people's questions and posts on this forum without taking any kind of side. or making any moral judgements. But I must say that I find your husband and your sister to be morally indefensible. You would have to be a truly horrible woman to deserve what they are doing to you, and I have to say that you don't come across as being horrible in the slightest. I know you face an uncertain future, whichever course of action you choose to take, but I do wish for your future to be much happier than your present.
At 44, you are still young, with plenty of years left ahead of you. I urge you not to waste them in a relationship where no trust can ever exist again. Maybe you can turn things around, and use this violation of your trust and love to shake things up in your own life. Perhaps it's time to make some changes to the way you feel, the way you look, and the way you let yourself be treated. When you feel happy about yourself, then the future doesn't seem quite so dark, and you can face it with hope rather than fear.