OofS,
It looks like you've done a good job of covering the bases, with regard to your friend's bitterness toward God.
What happened to her father was not God's fault. And He doesn't have anything to prove anything to any of us.
Still, God does love us all. He wants us to trust Him and turn to Him in our need. When we do this, His goodness, power and presence becomes obvious to us.
It is we who have much to prove to God. In life, we come continually to the fork in the road. There we choose to look to God, or we choose to turn from Him. We choose to place our faith in God, or we place our faith in ourselves, or in someone or something else. We love God above all else, or we love something else. We keep His commandments, or we do not. The choice - and the consequences for our choices - are always ours.
Please keep trying to help your friend, but realize that the decisions are her's alone to make. Pray for her, and support her however, whenever and wherever you can.
And you keep on trusting God. That may be the best thing you can do to help your friend. She is in a dark place. You are a light shining into that darkness. Make sure that the source of your light is God, and that it leads from her darkness to His Love.
God bless,
jimmy