Whatever about the rights and wrongs and legal liability, if she is your nanny and you appreciate her, then why not pay the medical expenses yourself as a gesture of goodwill?
Your policy, at best, covers your legal liability and it is difficult to see, on what your have posted, how you would be legally liable for her fall in a third party property. As PeteB says, if there is a defect/deleterious matter on the stairs that caused her to fall,l then that could put liability on the property owner. Slips and falls are difficult cases to win, on either side!