A Flight Service Station reported an aircraft with a failed engine diverted to a nearby airport; and expressed concern about the facility's communication capabilities.

Narrative

Aircraft X called up stating ZZZ radio; I advised this was ZZZ1 radio and to check freq. He called again asking if ZZZ1 radio was open and that he had a failed engine. Advised to request priority handling; say number on board; and that emergency crew would be ready. Gave current wind (twice) and altimeter. Stated no traffic. Being the aircraft was within 10nm and unsure of alt. I chose not to give AIRMETS to keep distraction and radio usage low; and believed in giving them they would not provide additional aid or new information that I'd already provided. Aircraft Y (company traffic) was in trail and provided visual assistance. Aircraft planned and landed rwy XX. No further assistance needed. However; during the event; I had issues with trying to call out due to line being u/s. First attempt center's phone line was busy; second time I was able to verify they had him on [Squawk] XXXX and location. Issues with trying to call personnel; having to dial X-XXX vs just XXX and having only one phone line available. Having to make multiple attempts dialing each phone number took time and focus away from tracking and providing better assistants to the aircraft.Recommendation: Fix line with center and possible have another line to call personnel; along with having a second phone line so if you are on the line with someone else you are still able to get another call in/out.