What do you do to atone for betraying your friend and revealing her secret that she entrusted you with, given your fear of God and your desire to avoid bad relations with her if you confess to her?

Disclosing a secret is forbidden unless there is an overriding benefit. You must repent and seek forgiveness for disclosing your friend's secret, which she entrusted to you. If possible, be direct with her and ask for her forgiveness; otherwise, generally ask for forgiveness and make an excuse about the difficulty of fulfilling the rights of companionship, and seek Allah's forgiveness for her.