The tenancy agreement may set out expressly what repairs the landlord has responsibility for, often the landlord is responsibility for repairs to the structure, the outside of the building and fixed items and the tenant responsibility for decoration and of course repairs to their own equipment, but this may not be the position in your case, so first establish who is responsible for the stairs and any other area which causes concern.
If it is found that the landlord is responsible he will not be liable for the repair unless you have given him notice of the need for repair. So if a tenant is injured (I note your partner was not injured) as a result of a defect the landlord will not be liable for the injury if he had not been notified of the need to repair the defect. Your first action should be to notify your landlord of the need to effect repairs to areas that he is responsible for and this should be done in writing, the notice does not have to be in any specific form and you should keep a copy. If a landlord fails to meet his obligations the tenant would have several remedies one of which would be to take court action, before this is undertaken I suggest you consult a solicitor.