Because Jo and Dr. Stadler had signed their divorce papers, she would no longer be considered next of kin for purposes of withholding medical treatment or determining his wishes on organ donation.
---- (This is incorrect. It would only be true if the papers had already been filed with the court; just signing the papers doesn't make a divorce official - it has to be legally filed with the court and officially ruled on by a Judge, and since in the show, it has only been about a 12 hours to a day since they signed the papers, they most likely wouldn't have been filed yet, making Jo still legally able to make the decision.)
Paul's breathing tube is not connected to the oxygen supply during the organ harvest surgery.