As a personal and professional practice, I like to retain copies of anything that bears my signature. Is there anything that I can do to compel him to produce a copy? a demand letter or some other vehicle?
What makes you think one human (acting legally) can compel another human to do anything?
Only a court can compel people to act or not act!
And, even at that, all a court can do in the end (should one choose not to obey the court's order) is hold you in contempt and fine you or lock you away for a few days!
The time to have ASKED for a copy was at the time you signed the lease.
But, if the landlord isn't willing to give you a copy, maybe he never kept a copy?
Either way, without a lease, you're a month to month tenant.
You can write the landlord a letter and ask him to allow you to make a copy of the document.
All you need is a smartphone with a camera or a digital camera.
Other than that, you'll have to haul him into court and request a judge provide you with relief.
However, the landlord will probably say you never had a lease.
Good luck.