The creation of Anne Boleyn, Susan Bordo, 126-7.
"The problems with theories that postulate a crisis that turned Henry from a virtuous prince into the sort of man who could order the execution of a wife is that Henry was always capable of decisively and irrevocably turning of the switch of affection, love, tender feeling, and shared memories; striking a fatal blow; and refusing to look back. In fact, those whom he loved most - Wolsey, More, Anne, Cromwell - were most at risk. Because he love them, they had the most power to disappoint him - and for Henry, disappointment could never be "slight". All wounds to his authority, his manhood and his trustwere bloody gashes that he could repair only by annihilating the one who inflicted the wound."
Henry had moved heaven and earth to get Anne as his queen, had alienated Europe, the pope, his own kingdom to a large degree and his most trusted advisors, had executed More, Fisher and many other. He had discarded his only surviving child and his most worthy wife to get a son with Anne. And what did he get in return? A girl baby. No, even if the charges were bullshit and I'm 99 % sure that they were, it in itself was irrelevant. She had failed to be what Henry had wanted her to become. She had failed him in the most horribly way. He almost died in 1536 and god do I wish he had, but the time had come to do away with Anne. The adultery was just a excuse at this point. If wasn't like Katherine Howard, at that point Henry had a son. I'm sure that Henry considered adultery a acceptable reason to axe a consort, ergo Katherine Howard, but at Anne's time the adultery was just the official reason.
Jane was a quiet simpering little dishrag who was Anne's opposite, so why deal with Anne when she had failed her purpose?
There is also a broad spectrum between doing nothing and publically executing a consort for the reasons you provided.