As Salamu Alaykum. I broke relations with my brother, who is not a Muslim, due to his wife and my husband flirting with each other. Before I took this measure, I explained to my sister in-law that we Muslims do not mix or talk to the opposite gender unless there is a need for it, yet my husband despite my advice kept talking to her freely and even ignoring my brother and my presence. I miss my brother and my family is sad for this because we have always been very close and kind to each other.
What can I do? My husband denies his misbehavior and blames and accuses me of jealousy. I feel he is more responsible for this sad situation because he is a Muslim and my sister in-law is not; therefore, he should establish the limits and act as a Muslim man should. It has been several years I do not speak to my brother and my entire family is sad and disappointed with the situation and started to blame Islam for the action I took of remaining apart to avoid more troubles.
Jazaka Allah Khair, Was-Salamu Alaykum wa Rahmatu Allah

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

If your husband behaves in the manner which you have described in the question, i.e. flirts with your brother’s wife, then he is, without doubt, negligent in observing the rulings of the Sharee’ah. Your brother's wife is not among the Mahaarim women of your husband. In fact, the prohibition is even more emphasized and his sin is graver because she is married. The fact that she is a non-Muslim does not constitute a valid justification for such an attitude on part of your husband.

We advise you to keep on advising him to repent from his sinful acts. You should make sure that you advise him kindly and wisely so that your advice will be acceptable and fruitful. You should also remind him that he has to be a good example for others in terms of highlighting the virtues of Islam concerning preserving people’s honor and refraining from violating it. Besides, you should tirelessly supplicate Allah to guide him to the straight path. Allah, The Exalted, has enjoined us to implore Him and promised to answer our supplications. Allah, The Exalted, says (what means): {And your Lord says, "Call upon Me; I will respond to you." Indeed, those who disdain My worship will enter Hell [rendered] contemptible.} [Quran 40:60]

However, you should not have severed your relationship with your brother. In fact, even if we assume that your brother is aware of his wife's inappropriate attitude, severing your relationship with him may not be beneficial. It may even urge them (your brother's wife and your husband) to be more stubborn. It may also reflect a distorted image of Muslims and Islam.

Hence, if you do not fear any potential harm in maintaining the ties of kinship with your brother, then do so and hope for the reward of Allah in return for your deed. You should show kindness to your family in general as this may encourage them to embrace Islam. Allah, The Exalted, says (what means): {Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them. Indeed, Allah loves those who act justly.} [Quran 60:8]

Moreover, Asmaa’ bint Abu Bakr reported: "My mother came to me during the lifetime of Messenger of Allah and she was a polytheist. I said to the Messenger of Allah (seeking his verdict), "My mother has come to me and she desires to receive a gift from me, shall I keep good relations with her?" The Prophet said, "Yes, keep good relation with your mother." [Al-Bukhari and Muslim]

We advise you to persistently supplicate Allah to guide your family to Islam and to grant them success in embracing it.

Allah Knows best.