Tell your letting agent he is confused.
It is only if a T is exercising a break clause within a fixed term AST that he can be required, contractually, to give more than one month's notice.
Your tenancy is now a statutory periodic tenancy ('month by month' or 'rolling'). The statutory notice period for that is only 1 month. (For your LL to give you notice however, it is 2). The agent/LL cannot override that legal right, whatever nonsense they choose to write into the contract.
As long as you have given at least one month's notice to end on the day before the day of the month on which the tenancy originally began (have you?), then your notice is valid and your rent obligation ceases on the day it expires. If you stay even one further night after that day, though, you will owe rent for the following month.
BTW, you can still be bound by a contract even after your fixed term ends - but only in the terms an conditions which do not relate to the end of the fixed term. All the other stuff (rent, repairs, etc) can remain valid.. If no new contract is signed, your tenancy becomes a SPT by default and the old contract terms still apply, by and large, but with the statutory provisions now in place re ending the tenancy.
'Pause you who read this, and think for a moment of the long chain of iron or gold, of thorns or flowers, that would never have bound you, but for the formation fo the first link on one memorable day'. Charles Dickens, Great Expectations