Oi Pessoal! :)
I've never heard of laxness. I would use laxity.
Due to his laxity, the bills went unpaid.
Because he was so lax in his responsibilities . . .
Oversight, as mentioned, sounds like a good equivalent also.
Due to the clerk's oversight, the company lost $1,000 in one day.
Negligence might also be used.
Due to the negligence of the security officer, the office was burglarized last night.
The security officer was negligent in his duties and got fired because of it.
Carelessness, again, as mentioned, would also be good to use.
To be careless indicates more of a general lack of attention; to be negligent is a little more "forte" in meaning. The word "negligence" can be used as a legal term, carelessness would not (though it may be contained in descriptions).
Just a few comments/suggestions,