Should Brown step down?

No he shouldn't – assuming there are no further significant facts yet to emerge.
If Mr Brown were to allow a kiss-and-tell story to lead to his resignation as mayor, then that would be to give in to the politics of the gutter. Len has admitted he's done wrong by his family. Any further concession would imply that it's open season for all jilted ex-lovers of prominent New Zealanders to let loose with their political and personal vendettas. But the country has more important matters to deal with. And Auckland has more important matters to deal with.
The story undoubtedly does Mr Brown some political damage, though. Aucklanders will perceive him differently for the time being. Many do feel his integrity and trustworthiness are in question. But all that can be restored in time, and the affair is not a sacking offence.
Winston Peters has waded into the argument, asking why the right-wing blogger didn't release the story before the election so that voters could pass judgement accordingly. Well, that's one way to increase voter turn-out! But, seriously, the answer to Mr Peters' question is simple: the woman concerned, Bevan Chuang, was herself a candidate for a post on a local board. She surely would not have allowed explicit revelations about her own sexual conduct to affect her electoral chances.
Ms Chuang was a 'Communities and Residents' (centre-right) candidate for one of 4 members on the Maungawhau Subdivision of the Albert-Eden Local Board. Three other C&R candidates were successful. But, in a close-run ballot, Ms Chuang was pipped at the post by a rival from the centre-left 'City Vision' ticket.
What I would be interested to know is whether Ms Chuang would have withheld her X-rated gossip affidavit had she been successful in the local-board election...? But I guess we'll never know the truth about that.

Update: News in the NZ Herald about Ms Chuang and her claim that she felt under pressure from others to reveal details of her liaison have overtaken (and, I argue, vindicated) the above post. It also supports my earlier point about why we should not publicly reveal such matters in the first place. The unwritten 'rule' about not reporting on the private lives of public figures should be reinforced by this whole saga.

A further update: The later report about Mr Brown acting as a referee for Ms Chuang and the announcement of an inquiry into these events by the Council look like the kinds of 'significant facts' I referred to above in this post. So, I will now have to reserve further comment.