I find many people are like your friend: They don't have a hard time believing in God, but they do have a hard time trusting Him and putting their lives into His hands.
Sometimes they feel this way because–like your friend–they've had some bad experiences in life, and these have made it difficult for them to trust anyone (including God). Others, however, resist trusting God because they don't want to lose control of their lives. They know God wants them to hand their lives over to Christ, and they don't want to do this. Instead, they want to keep running their own lives.
Pray for your friend, that God will help her overcome her fears and doubts. In addition, ask God to help you explain to her that God isn't like her father, or like anyone else who might have hurt her. God loves her, and He can be trusted–completely and without reservation. Other people may fail us, but He never will. God's promise is true: "Never will I leave you; never will I forsake you" (Hebrews 13:5).
How do we know this? We know this because Jesus Christ was willing to leave heaven's glory and die on the cross for our salvation. Point your friend to Christ, for He alone is worthy of her complete trust and love. Is your trust in Him?