I don't subscribe to the idea that all sins are equal. Cheating on a woman you barely know is considerably less treacherous than murdering a king as dishonorably as possible or slaughtering women and children. While Ned did indeed break his vow and stain his honor, I'd arguing that bearing that blight and raising Jon Snow with as much love as his legitimate children, instead of casting his poor bastard aside like so many other men had and would have done so as not to sully their good name, is an even greater testament to his honor.
Had Tywin taken King's Landing with more finesse, and say, secured the Mad King's family in their quarters, I feel like Ned would have been more willing to hear Jaime's account of Aerys's death.