Thank you for your question and patience, I’m Tom and I’ll try to help you.
Unfortuantely, your position is not good. If it is the case that your periodic tenancy runs from 5th of a month to the 4th of the next month then legally you can only end the tenancy by serving one month’s notice with such notice to expire at the end of the rent period on the 4th of a given month.
The landlord is under no legal obligation to accept any lessor notice period or other form of compromised monetary payment to accept that the tenancy is ended earlier.
Your only legal right here is to serve a correct notice which is a month in length and ends of the 4th of a month, which means that the earliest notice you could now serve would give a notice period of 5th December to 4th January and you would have to pay the rent for this period.
If you did not give this notice or did not pay the rent for the notice period then the landlord would have the option of suing you at county court for a money judgement on the rent that you would have paid throughout the correct notice period. This would be a CCJ and it would affect your credit rating, so you should avoid this.
I know this is not the answer that you wanted, but it’s important that you know your correct legal position so that you do not prejudice yourself further.
My goal is to provide you with a good service. If you feel you have received anything less, please reply back as I am happy to address follow-up issues specifically relating to your question.