Hence why I only do month to month leases anymore, such as when I signed my most recent leases/renewals. With annual leases you take on all of the risk. I'd rather have the unit vacant than a crappy tenant. I'd give 20 days notice (in WA) to vacate, then once they are moved out, take it out of their security deposit. This way they are out of my property so they cannot do more damage out of spite, and if they come back and do damage to the exterior, I'll involve the Police.
I think there is a clause in my leases about Attorneys fees if I win, in that they have to pay mine, should they bring an action against me. So this way I can easily and quickly get rid of them, take the money out of their security deposit, and make it to my advantage if they try to bring legal action against me.
Oh, by the way, I had a tenant recently try to become a princess on me after moving in. After I caught her in several lies, I gave her the option above and she straightened out right away and the BS stopped.
The way I see it, you are now forced to replace the door rather than just the lock, as a result of damages that he is obviously lying about, with pictures to prove it. I mean, he doesn't deny that damage has occurred on the count of the evidence probably, he just blames it on his kids, which makes no difference, not that that obvious lie holds any water. Even if it did, he is still responsible for his "kids" attempt at playing Bob Villa.
Since you are stuck in a lease, you may not have it as easy as my suggestion, but I strongly urge you to only do month to month leases moving forward.