We are truly sorry to hear about the serious health issues your dog is facing and the emotional turmoil you're experiencing. Firstly, we want to address the fact that you mentioned you are not religious. We understand the importance of prayer to Christians and the scripture says that without faith it is impossible to please God. If we are to pray for you and you do not believe then it is impossible for us to help. It is only by the name of Jesus that we have any access to God. It is also only by that name that anyone can be saved.
For there is one God, and one mediator between God and men, the man Christ Jesus. 1 Timothy 2:5 WEB
And in none other is there salvation, for neither is there any other name under heaven, that is given among men, wherein we must be saved. Acts 4:12 WEB
That being said, we will pray for you. While we pray, we want to remind you that God cares for you and your situation. He even cares for your dog's suffering. In the book of Matthew, it is written:
Look at the birds of the sky: they don't sow, neither do they reap, nor gather into barns. Your heavenly Father feeds them. Aren't you of much more value than they? Matthew 6:26 WEB
God values you greatly and wants to hear from you. Turn to Him in this time of need, and cast your anxieties on Him, because He cares for you.
In all anxiety, direct your requests to God in prayer and petition with thanksgiving and trade your anxiety for the peace of God that surpasses all understanding, which will guard your hearts and minds in Christ Jesus. Philippians 4:6-7
But seek first God's Kingdom, and his righteousness; and all these things will be given to you as well. Matthew 6:33 WEB
Regarding your mother, it's important to remember that people deal with difficult situations in different ways. It's possible that she cares more than you realize, but she might be struggling to express it.
The scripture says
If it is possible, as much as it is up to you, be at peace with all men. Romans 12:18 WEB
We will pray for her and for your relationship, that God may bring comfort, understanding, and unity during this challenging time. God will provide you with strength to go through this.
We lift up your dog to the Lord, asking for His healing touch and mercy. We pray for provision, that God may supply the needs for medication and treatment. We also pray for your comfort and peace, trusting that God will carry you through this trial.
Father, we come to you in Jesus name. We bring this lost and hurting individual to you. We ask that you would be with them and help them to believe in you and your son Jesus Christ. We ask for your grace and mercy during this difficult time. We pray for healing for their dog and for provision for the needed medication. We pray for comfort and peace for their heart, and we ask that you would draw them closer to you. We come to you in Jesus name. Amen.
To know that people care, we encourage you to reach out to local churches, Christian communities, or other believers who can offer support and prayers for you and your dog. God's people are called to bear one another's burdens, and they may be able to provide the encouragement and help you need.
Finally, we want to gently remind you that our hope and love should ultimately be rooted in Jesus Christ, who never leaves us nor forsakes us. While the love and companionship of a pet are truly special, only Jesus can fill the deepest needs of our hearts. We pray that through this experience, you will draw closer to Him and find comfort in His unfailing love.
Prayer Focus: God, Thank You for loving me. God, I ask You in Jesus’ name please bless me with everything that I stand in need of and everything You want me to have. 
