Not surprisingly a number of questions about security deposits are always in the mix in the New York landlord and tenant questions and answer. That is due to the fact that New York State (almost alone in the entire country) does not spell out step by step any method at all for a landlord to return security, retain security, or keep a part of the security deposit.
Basically, New York State forces every tenant with a security deposit issue into suing in a local Small Claims Court if after a "reasonable period" the landlord did not return the security deposited at the start of the tenancy or does not inform the tenant of charges made to the security deposit, or returns a small portion or none at all for alleged damages. New York does not require a walk through of the apartment or any other step to properly return the security deposit to the tenant. New York simply leaves a landlord and a tenant the right to make up their own terms about the use of the security and when the deposit is returned.
Forget about giving notice by Email as you are unlikely to convince an arbitrator or a judge that Email notice was allowed by the terms of your expired lease. Likely, your lease required that any notice sent to the landlord by the tenant was to be mailed to the address of the landlord as provided in the lease for notices, and mailed either by certified mail or registered mail. You have to verify those details from your own lease.
You have to provide the landlord your new mailing address.
Hopefully you documented the apartment condition on your move out date.
If the landlord followed the law, and truly placed your money ( a security deposit is the tenant's money held in trust to be returned at the end of the tenancy pursuant to the terms of the lease) into a bank in a security deposit account, then all the delay so far may be attributed to the slow process that New York banks go through when giving up control of any money in a deposit account. The bank may need to wait until the end of an interest earning cycle to cut the check. The bank should be issuing the check in your name, and would ordinarily mail your check to the landlord. If the landlord did not invade the funds on deposit for "repairs" then you should receive the entire amount along with a tiny amount of earned interest.
Because New York does not provide in a clear and simple statute what a reasonable time is courts run the gamut from four weeks to 45 days, but that really means nothing, if the landlord does not have from you a proper written notice presenting your new address.
So start by sending the letter.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.