His biggest problem is that he's made it clear to all that he's a pathological liar. A liar willing to say anything to save his ass. Now, I know that this trait endears him to progressives, but fortunately they're not running the country. There's nothing the man can say about anything at this point that will be believable and that makes him totally ineffective at his job. He can't be believed, he can't be trusted.
A man with scruples, a man with a sense of decency, a man with a moral compass would resign. Mr. Weiner obviously has none of these traits. I fully believe him when he says he will not resign.
He's asked for a leave and his fellow representatives will have to vote on that request. It's pretty clear based on what his peers are saying publicly that he'll be denied that leave. He's boxed himself into a corner, but it's the only option he had left.
He's trash and he belongs at the curb.