No matter how hard we try or how many arguments we present, ultimately only God can persuade a person that the Gospel is true. Jesus said, "No one can come to me unless the Father who sent me draws him" (John 6:44).
This is why the most important thing you can do for your neighbor is to pray for her. Right now, she doesn't believe she has any need to have her sins forgiven - and therefore she doesn't believe she needs Christ. She takes pride in her goodness - but her pride is like a barrier that must be torn down. Pray that she will face her own spiritual emptiness, and that God will convict her of her self-righteousness and her need for Christ. The Bible says the Holy Spirit "will convict the world of guilt in regard to sin and righteousness and judgment" (John 16:8).
Then ask God to help you be a witness to her by the way you live - including the way you react when things don't go the way you wish they would. Anyone can be at peace when things are going well - but how do you react when things turn against you? The Bible says, "But the fruit of the Spirit is love, joy, peace, patience, kindness, goodness" (Galatians 5:22). Are these evident in your life?
God has given this neighbor to you; she isn't there by chance. Ask Him to help you be a blessing to her by pointing her to Christ, who alone can give her hope for this life and the next.