ZLC Controller reported chronic communication line failure to Control Tower resulted in aircraft entering a higher MVA and a CFTT event.

Narrative

Aircraft X on RNAV31 approach to KUN. Aircraft lost GPS capability for the approach crossing JUNOL; still on a NE heading past the turn at JUNOL and headed toward the mountains the run along the east side of SUN. Tower called and informed that the pilot had a GPS problem and couldn't continue the approach; and they needed a heading and altitude to issue. At this point the aircraft was at 086 in a 100 MIA; and headed toward a mountain range with a 109 MIA. I issued left 160 heading and 120 altitude. I was fully aware that the aircraft was below the MIA; but considering the situation (IMC with a loss of gps navigation and the proximity to even higher terrain and mountains); I felt it was an emergency situation that required immediately getting the aircraft turned away from the mountains while climbing. After giving the tower time to issue the clearance; I called them back (this took much longer than normal because the shout line to the tower was partially out of service; so after trying; I had to call the phone number instead) and asked them to issue a low altitude alert to the aircraft who was still well below the terrain in a 100 MIA. By the time this was relayed to the tower controller they had already shipped the aircraft back to me and they were climbing out of 100 clearing the MIA; so the low altitude alert was never relayed to the pilot.Fix our communication lines. The shared TWF/SUN shout line has regularly had problems and outages over the past couple years; much more than other lines to other facilities; and it's the one we use the most so it is a major problem. This line needs to be fixed properly so this doesn't continue to be a recurring issue in the area. Although that wouldn't have changed the instructions to the pilot which was relayed over the shout line when tower called us; it would have allowed for a timely issuance of a low altitude alert to the pilot.