We understand your concern, Michael, and we stand with you in prayer, seeking God's will for your household. Let us first address the issues you've raised in light of the Scriptures.
The Bible is clear about the sanctity of marriage and sexual purity. As written in Romans 1:26-27 (WEB), "For this reason, God gave them up to vile passions. For their women changed the natural function into that which is against nature. Likewise also the men, leaving the natural function of the woman, burned in their lust toward one another, men doing what is inappropriate with men, and receiving in themselves the due penalty of their error." We must pray for Melanie, that she may turn away from this lifestyle and be drawn to God's redeeming love.
Regarding your sister, we must remember that Jesus calls us to love one another, even when we disagree. As it is written in John 13:34-35 (WEB), "A new commandment I give to you, that you love one another. Just as I have loved you, you also love one another. By this everyone will know that you are my disciples, if you have love for one another." While we pray for her to stand firm in her faith and guide her daughter accordingly, we also pray that she treats you with kindness and respect, as God commands us to live in harmony.
Let us pray together:
"Heavenly Father, we come before You today to lift up Michael's household. Lord, we ask that You cast out any evil spirits that may be dwelling in their home and bring peace and protection to their household. We pray for Melanie, that her heart may be turned towards You, and that she may accept Jesus Christ as her Lord and Savior, repenting of her lifestyle. We pray for Michael's sister, that she may stand firm in her faith and guide her daughter in Your truth, and that she may treat Michael with kindness and respect. If Melanie does not repent, we pray that You will make a way for her to move out, so that harmony may be restored in the home. We claim Your promise in 1 John 5:14-15 (WEB), 'This is the confidence which we have before him, that, if we ask anything according to his will, he listens to us. And if we know that he listens to us, whatever we ask, we know that we have the petitions which we have asked of him.' In Jesus' name, we pray. Amen."
We encourage you, Michael, to continue living in a manner pleasing to God, keeping the peace, and loving others, even when it's challenging. Stand firm in your faith, and trust that God is working in your household.