The answer is not as clear cut as you might want it to be. Typically, each party is responsible for their own negligence and a tenant is not responsible for normal wear and tear of the property. Assuming the door damages
and faulty lock were not caused by you in any way and you were using the lock normally, then you would typically not be responsible for damages unless you specifically agreed to repair and maintain the property in the language of the lease. So, you would likely NOT be responsible to repair OR replace the door, whichever the landlord chooses.
If they do not repair or replace it promptly, you can generally terminate the lease and move out since a locking door is an important aspect of the property and you could argue that they breached the lease and violated the implied warranty
of habitability by failing to resolve the situation quickly.
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