The Arizona pollitical conundrum
Kyl also said he has committed to serving "through at least the second session of the 115th Congress," i.e., the current session of Congress, which ends on Jan. 3. That abbreviated timeframe would nevertheless give him the chance to vote to confirm Brett Kavanaugh to the Supreme Court—a process for which Kyl rather conveniently has been Kavanaugh's so-called "sherpa."
Whether Kyl sticks around beyond that may depend on Ducey's fate. Ducey faces a competitive re-election challenge this year from Democrat David Garcia, so if Ducey loses, that would probably incline Kyl to stay put lest Garcia get to pick a replacement. (Under Arizona law, Garcia would be required to pick someone "of the same political party as the person vacating the office," but the statute offers no specifics beyond that, so he could choose a Republican in name only.) Of course, in that scenario, Kyl could bail just before Garcia were to be sworn in, giving Ducey a second crack.
But why the short commitment from Kyl in the first place? It's hard to say, but here's a theory: If GOP Rep. Martha McSally, who is running for Kyl's old Senate seat (now open thanks to Sen. Jeff Flake's retirement), loses this fall to Democrat Kyrsten Sinema, Ducey could then appoint her in Kyl's stead. McSally would then be able to run in the special election in 2020 as the incumbent. We'll just have to see what happens in November.