What should I do upon discovering my wife's betrayal for over 3 years with my closest brother, and the resulting abomination of adultery between them, and how should I act to preserve the reputation of the three families in light of this shocking situation?

If the wife repents from the sin, then cover her fault and keep her, while being cautious in the future about any non-mahram entering upon her. However, if you are not sure of her repentance, then divorce her kindly and cover her fault, and use tawriya (ambiguity/equivocation) if asked about the reason for the divorce, in order to protect her reputation.