You now have a month to month tenancy in the substitute apt. If you wanted to terminate that tenancy you must give 30 days notice. If the LL did not place you into the apt you signed a lease, paid rent and security you may sue for 2x the rent for not placing you into possession. You have a choice stay and wait or find another place give notice and demand your money back or file suit.
your LL is in breach of your lease agreement on the initial apartment, and you are probably entitled to some damages as described by Ms. Joseph. In addition, your substitute apartment appears to suffer from serious defects with regard to habitability which may entitle you to damages, and/or may be sufficient to provide you with an "out" entirely as a constructive eviction.
You should speak with an attorney to discuss your specific situation
The foregoing is not legal advice, and nothing in the foregoing shall be deemed to create an attorney client relationship. If you feel you need to speak with an attorney regarding your issue, it is recommended that you contact an attorney with expertise in your area of inquiry. The information related above is purely for informational purposes, and should not be acted upon without speaking with qualified counsel familiar with you specific situation and the laws related thereto.