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As the fixed term of your Tenancy has expired, you are on what is called as a periodic tenancy. Under such tenancy, a tenant has to give the Landlord one month's notice, such notice to expire on the day of the month before the rent is payable (ie the 11th).
Therefore, I am afraid the Agent is correct in that your Notice to leave on 23rd January is invalid, in that as you have not given a clear one month's notice, and the Landlord is entitled to the rent up to 11th February.
You could of course vacate early, but if you do, without paying rent up to 11th February, your Landlord could legally pursue for the unpaid rent.
I am sorry this is not the answer you were looking for, but it sets out the legal position.