We understand that you are going through a challenging time with your father's health and the complex family dynamics at play. Let's bring your situation before the Lord in prayer and seek His guidance and intervention.
Firstly, we need to acknowledge that healing and breakthroughs come from God alone. It is by the name of Jesus Christ that we have access to God. As it is written in Acts 4:12 WEB: "There is salvation in none other, for neither is there any other name under heaven, that is given among men, by which we must be saved." Let's ensure we are praying in His name and seeking His will.
Let's pray together:
Dear Heavenly Father,
We come before You in the name of Jesus Christ, lifting up our brother who is in desperate need of Your intervention. Lord, You know the situation with his father, who is in the hospital, facing a serious health condition. We pray for Your healing touch upon his father, that You would restore his health, his ability to eat, drink, speak, and walk. Lord, guide the doctors and medical professionals, give them wisdom and clarity in their diagnosis and treatment. We pray for a competent neurosurgeon to be available for Your servant's father, Lord provide the finances for the surgery if needed.
Lord, we also pray for unity and understanding among the family. We rebuke the spirit of arrogance and negativity that is causing division and hindering the family from coming together in faith and support. We pray for a heart change for the brother and mother, that they would be open to faith and hope, and that they would support the efforts to seek help for their father and husband.
Lord, give our brother the strength to endure this resistance from all sides, and the wisdom to navigate this challenging situation. Guide him to the right people, the right doctors, and the right resources that can help his father in this time of need.
We stand in agreement with our brother, believing and trusting in Your power to heal and restore. We claim the promise in Matthew 18:19 WEB: "Again, assuredly I tell you, that if two of you will agree on earth concerning anything that they will ask, it will be done for them by my Father who is in heaven.", and in Mark 11:24 WEB: "Therefore I tell you, all things whatever you pray and ask for, believe that you have received them, and you shall have them."
We ask all these things in the mighty name of Jesus Christ, our Lord and Savior. Amen.
Now, let's address some points from your request. It's important to remember that our faith and trust should be in God, not in man or connections. As it is written in Psalm 118:8-9 WEB: "It is better to take refuge in Yahweh, than to put confidence in man. It is better to take refuge in Yahweh, than to put confidence in princes.", and in Jeremiah 17:5 WEB: "Thus says Yahweh: Cursed is the man who trusts in man, and makes flesh his arm, and whose heart departs from Yahweh."
However, this does not mean that we should not seek help from others. God can and often does work through people to accomplish His purposes. But our ultimate trust and dependence should be on God alone.
Regarding your mother's negativity and resistance to faith, remember that we cannot force anyone to believe, but we can pray for them and be a living testimony of God's love and power. As it is written in 1 Peter 3:1-2 WEB: "In like manner, wives, be in subjection to your own husbands; so that, even if any don’t obey the Word, they may be won by the behavior of their wives without a word; seeing your pure behavior in fear.", and in Romans 12:18 WEB: "If it is possible, as much as it is up to you, be at peace with all men."
Lastly, let's remember that God is sovereign and His ways are higher than ours. As it is written in Isaiah 55:8-9 WEB: "'For my thoughts are not your thoughts, neither are your ways my ways,' says Yahweh. 'For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts.'"
Keep trusting in God, keep praying, and keep seeking His will. He is faithful and He will see you through this challenging time.
We are standing with you in prayer and belief, trusting God for a breakthrough in your father's health and your family's unity.
Prayer Focus: God, Thank You. Thank You for loving me. God, I ask You in Jesus’ name bless me with everything that I stand in need of and everything You want me to have. Bless me to know You in truth, fall in love with You with all my heart, mind, soul, body, and strength and never fall out of love with You. Bless me to have an ever growing closer stronger, more intimate relationship with You. Bless me with the love, desire, strength, and the spirit of obedience to always delight myself in You, seek first Your kingdom, and Your righteousness. Help and strengthen me God to always respect and obey You. Bless me to trust You with all my heart, acknowledge You in all my ways, and lean not to my own understanding. 
