A Center Controller reported a Fractional Jet with a failed transponder descended and diverted to a nearby airport.

Narrative

I was working Location A and the Location B sector next to me lost their frequencies. I put in my settings that cover both sectors and started bringing up data blocks. Through adjacent sectors and facilities; CPDLC and guard we were able to get the 4 aircraft on my frequency. I had about 8-10 aircraft between the 2 sectors. However we had a thin line of thunderstorms in the middle of the 2 sectors. Tops were between FL350-FL400 so we add deviations and altitudes changes taking place to make it through.Aircraft X was just east of ZZZ AT FL390 and requested FL350. I descended the aircraft and the last hit I observed was at around FL360. I continued to do other things and noticed the altitude in the data block change to xxxx. I watched it for an update because it was in an area with overlapping radar coverage that sometimes causes CST (Coast) tracks and ghosting. When it didn't change after the update I informed the aircraft that I was not receiving their transponder and reassigned their beacon code. I did not receive a reply from the aircraft. I was about 30 seconds and tried again and got no reply. I informed the Supervisor about what was happening. I tried a 3rd time and shortly after the aircraft finally called and informed me that they were [requesting priority handling] and needed to descend to 14;000. I had already pointed out the aircraft to the sectors below me and yelled over that the aircraft was descending to 14;000. I was on my super high setting; so I jumped over to my lower setting with altimeters and gave the descent and altimeter and got an altitude report (FL298). The aircraft called back a minute or so later and asked for an airport to divert to. I gave [them] the location of ZZZ1 airport; and [they said they could accept that and I got another altitude check around FL210. I descended the aircraft to 11;000 and our low sector contacted ZZZ approach who controls ZZZ1 and informed them and by this time a D side had been recalled to the area and he coordinated with all the in house sectors around us. The pilot asked for the ZZZ1 weather and my D Side informed me the next sector said they could issue it. I asked the pilot if [they] could accept a frequency change and they said yes so I switched them to the next sector. Several minutes late I was offered a break and took it. When I returned from break I was informed that I handled everything great but I forgot to clear the aircraft to the ZZZ1 airport. So I was going to receive a performance report. I started to work a sector but then asked to be taken off position because the report was distracting me and I was exhausted from the 2 sessions and I wasn't sure with the weather in the area if any happened that management would protect me or throw me under the bus.Recommendation: A D side probably should have been left at Location A after Location B sector had to be closed.