Cylinder lock help needed

by David
(Ajax, Ontario, Canada)

Patio door handle

After a bit of research, I've found that the easiest way was to put a locking cylinder on my patio door, accomplishing this by knocking out the punch-out on my surface mount patio door handle and inserting a cylinder lock.

The hole (punch-out) is roughly 15mm or 9/16 I believe.
I have a recessed mortise lock assembly.
Patio door handle is 3 15/16 center to center (4").
Patio door locking mechanism is single point, hook design.
Attached is a picture of the inside of the door handle...
I need assistance ascertaining the correct cylinder to use,
or would it be best to purchase a new pull ?
Warmest regards,
David O