Sure you can forgive the cheater without their repenting of their actions. In many cases, that does occur. Even when that happens, you will still need to allow them to go through the repentance process. Allowing them to repent is important. Repentance allows them to ‘wake up’ to the reality of what they have done. They need to wake up and be honest with themselves and with you about what happened. They also need to break through their bubble of selfishness and see how their actions hurt others rather than how their emotional needs are more important than others. The process of repentance allows them to achieve this. When you reconcile prior to repentance, it often short-circuits this process. The cheater will not want to repent if they do not have to. They do not want to face the tears and pain that often go with it.
I know that you do not want to see your spouse in pain. In the case of repentance, the pain is part of the healing process. When you want your relationship healed, you will also need to face these pains.
Nothing in this Work is intended to replace common sense, legal, medical or other professional advice. If your situation warrants it, please seek competent professional counsel.