A Tower Controller reported a departing Cessna reported a rough running engine and returned to land. The Controller allowed another aircraft to make an approach before they had permitted a runway inspection from Ground Personnel.

Narrative

Shortly after Aircraft X departed Runway XXL; I received a transmission without a callsign [requested priority handling] on XXX.XX. Aircraft X was my only aircraft on that frequency at the time; so I responded to them asking their intentions. When they said that they were trying to return with a rough running engine; I cleared them to land on their choice of Runway YYR or XXL and the Ground Controller called an ARFF Alert. There was no other traffic inbound to either runway. Aircraft X was able to gain altitude; enter a left downwind; and land Runway XXL. The [request for priority handling] occurred at about XA58Z; ZZZ Tower closes at XB00Z. I withheld making the closing announcement until Aircraft X landed at about XB05Z; as I wanted them to focus on landing the aircraft instead of worrying what services were being provided.As Aircraft X was in the downwind; I noticed Aircraft Y being vectored by ZZZ TRACON inbound to ZZZ [Airport]. I notified the Approach Controller about the situation and they broke the Aircraft Y off the approach. After Aircraft X landed; I started to call ZZZ TRACON to update them; but then heard an airport operations vehicle going out for a runway inspection. There was some confusion between myself; the Approach Controller; and the airport operations vehicle as we tried to sort out the situation; but it ended with Aircraft Y going around on about a 4 mile final and returning after the inspection was complete. I had forgotten that the [priority handling] services LOA requires a runway inspection for all Alert calls and hadn't expected it to be necessary for a rough running engine.