Center Controller reported an aircraft descended from its assigned altitude causing a conflict with another aircraft.

Narrative

I was controlling when I received a transmission from Aircraft X indicating that he wanted a descent. At the time; I had traffic 1;000 feet below Aircraft X and less than 5 miles separated. Consequently; I was unable to issue a descent clearance due to potential conflict. I informed the pilot of Aircraft X that the traffic was off his left wing and that I could not issue a descent clearance. The pilot acknowledged but insisted on an immediate descent. To mitigate the situation; I instructed Aircraft Y to turn 20 degrees right and then issued a 20-degree right turn to Aircraft X. Upon the next radar hit; Aircraft X was observed at 15;500 feet and descending; which triggered a conflict alert. The aircraft were not 5 miles apart at that point. I immediately asked the pilot of Aircraft X to state his intentions; and he reported a cracked windshield and the necessity to descend to 10;000 feet.Recognizing the severity of the situation; I [gave priority handling] to Aircraft X and cleared the aircraft direct to MYNN. Once Aircraft Y was clear of Aircraft X; I issued a descent clearance to 10;000 feet. I then coordinated with MYNN Control regarding the aircraft inbound and handed Aircraft X off to MYNN Approach. If the pilot would have stated the issue on the first transmission; I may have been able to expedite the entire situation. Instead; I was informed of the severity of the issue after the comfit had happened.