Must I confess my sin to my friend whose trust I betrayed, or is it sufficient to seek forgiveness and supplicate for her in secret?

Disclosing a secret is prohibited because it constitutes a betrayal of trust. If it leads to harm, then it is forbidden, as Al-Ghazali said: "Disclosing a secret is a betrayal, and it is forbidden if it causes harm, and it is ignoble if it does not cause harm." The author of Al-Insaf said: "Disclosing a secret is forbidden." Al-Mawardi said: "A man revealing another's secret is more heinous than revealing his own."

However, if no harm results from it, or if there is an overriding benefit in it, then there is no objection to it, just as Bilal (may Allah be pleased with him) did with the Prophet (peace be upon him).

Regarding confessing to your friend: Be open with her and ask for her forgiveness, unless doing so would lead to a greater harm. In that case, ask for her general forgiveness, citing the difficulty of fully fulfilling the rights of companionship. If she accepts, praise Allah; otherwise, seek Allah's forgiveness for her.