Holding the tenant liable for occupying the premises after moving out

I would like to know what is the ruling with regards to the following situation:

I have a flat which I let out. My tenant decided to move to another premises and gave me notice on the first of December for end of December. He paid rental for December already.

I clearly told him that kindly return my flat to me in the same condition which I gave it to you.

He said that we must do the repairs etc. and deduct it off the holding deposit which we have of his. We informed him that we are unable to repair the flat , because it is December and no one is around to work or supervise!

The tenant moved out of the flat during December, however left one of his cars parked at the premises and also some goods of his still in the flat as well as he’s ac. He didn’t handover the keys to myself, however he did give one key to the caretaker.

Fast forward to second of January, I’ve contacted him to remove he’s items and car And because of his items being in the flat and also being December and him not handing over the full set of keys to me, I was unable to repair and let out the flat to someone else.

Should the tenant be liable to pay rent for January? As he only removed the balance of his items and handed over the keys to me in January and after I called him. And we are only able to do repairs to the flat now and let out for for February