Can I sue for landlord negligence

Yes- you can sue your landlord, assuming you didn't agree in writing to assume responsiiblity for maintenance and repair of the stairs in question.  Your landlord is strictly liable for maintenance and upkeep of the premises, and had notice of the condition, and therefore an opportunity to repair, and failed to do so.  Unless for some reason your landlord tried to make repairs and you actively prevented the same (you don't say whether the stairs are indoors or outdoors) by failing to give access to make the repairs, which is your obligation, you can and should take legal action for money damages for pain and suffering for your injury and any lost wages.