I think there is a general misunderstanding about forgiveness and reconciliation in non-covenantal relationships such as friendships.
First, a few general things about forgiveness. When sinned against, the Christian has the obligation to forgive the person who sinned against them. This is basic Christianity. We forgive others because of the great sin debt of which we have been forgiven (Ephesians 4:31-32). It does not mean we approach the person who sinned against us and say, "Oh, I want you to know I forgive you for how you sinned against me."
The primary reason we do not do that is because they have not asked! They have not confessed their sin to you, they have not admitted guilt, they have not asked for your forgiveness.
However, this does not relieve you of your duty to forgive them from the heart. This means you must spend time with the Lord addressing the hurt or offense. You must repent of any anger, bitterness, or malice within you and come to a place of forgiveness before God toward the person who hurt you. This is not only responding in a Christ-like manner (and a command), but also your only defense against developing bitterness. You want to be ready to grant them forgiveness on the day that person comes to you in confession and repentance and asks you for it.
Forgiveness does not mean an automatic return to the relationship. Forgiving someone who has hurt you does not mean immediate reconciliation takes place. While the goal in broken relationships is always reconciliation, it is not always possible right away. Sadly some relationships are never reconciled to the place they were before the schism happened, if they are reconciled at all.
Friendship is a sacred trust whereby we invest our lives in each other. Women in particular share deep parts of themselves with other women in whom they place the trust of friendship, and these are not easily given once that trust has been shattered. If trust has been badly broken due to repeated offenses, even though forgiveness is granted it may take a significant amount of time for the offender to demonstrate repentance and rebuild trust. The person who is truly repentant will understand the gravity of their offenses and willingly give the necessary time to the person they have hurt. All sin has consequences, and time and distance apart may be a consequence of the sin they have committed against their brother or sister in Christ. This is not punitive, but a legitimate response to being hurt and exercising precaution against it happening again.
Words of contrition are often not enough to bring about reconciliation, especially if there has been a pattern of offense in the relationship. If you are wondering when to reconcile with someone, you would be wise to watch the life and conduct of that person to see if their life matches up with their words of repentance. If they resort to manipulation in an attempt to force you to reconcile with them you have good reason to be wary of their true repentance and change. Don't be guilted into a renewed relationship with them on the basis of their emotional needs or accusations of being unforgiving. You are being cautious and wise. Don't be a peace faker under the guise of unity in the church.
Be prayerful during this process. Pray for the one who hurt you, that he or she may truly be repentant and change. Pray the Lord would guard your heart as you observe the fruit being produced in his or her life. Be careful not to become a prideful fruit inspector, continually raising the bar and making it impossible for the person who hurt you to ever provide enough proof of change for reconciliation.
In the meantime, let them know you can forgive them and not be their friend. They may or may not accept that, and that is between them and God. Your responsibility is to be forgiving, watch their lives for evidence of repentance and change, pray for them, and trust that God is doing a work in both of you.