Your mother's landlord is not required to provide her with a temporary apartment while hers is being repaired. She should contact building management about getting access to remove belongings. She should also immediately go to Housing Court and initiate a case called an HP Proceeding against the landlord in order to get repairs made to the apartment as expeditiously as possible.
If your mother is a rent stabilized tenant she should file a reduction in services complaint with the New York State Division of Housing and Community Renewal. This will protect her right to return to the apartment once it is repaired and reduce her rent obligation to $1.00 per month during the period of time she is unable to live in the apartment. You can find the complaint form here:
The above constitutes general information only and should not be considered legal advice.
While I am not necessarily disagreeing with Mr. Bianchi, I want to point out that unless/until an attorney reviews your mother's lease, it's impossible to know for sure whether the landlord is obligated to put her up (unlikely but possible).
I'm just 3 "helpful" answers away from a free toaster-oven! I may be guessing or not licensed in your state. No atty/client relationship exists.