I'm thankful for your concern for those confined to nursing homes; many of them are very lonely, and your visits are probably the high point of their week. The Bible says, "Religion that is pure and undefiled before God, the Father, is this: to visit orphans and widows in their affliction" (James 1:27, ESV).
The most important thing you can tell this woman is to remind her that Jesus has already paid the price for all her sins—not just part of them, but all of them. When He died on the cross, all our sins were placed on Him, and He took the penalty we deserve. The Bible says that "in him we have redemption through his blood, the forgiveness of sins, in accordance with the riches of God's grace" (Ephesians 1:7). This is God's promise to her—and also to you, if you have committed your life to Him.
Satan wants to make us think our salvation depends on ourselves and how good we have been. But we can never be good enough—and that is why we need Christ. God's promise is clear (and He cannot lie): "He who has the Son has (eternal) life; he who does not have the Son of God does not have life" (1 John 5:12).
Pray with your new friend, and encourage her to put her trust in Jesus alone for her salvation. Encourage her also to thank God daily for saving her; a thankful spirit casts out doubt.