Were you phred in a prior life?

Never mind. Forum inside joke.

The reason I suggested what I did was to minimize travel time just getting to the shinkansen (bullet train) station to begin your rail trips on both days. The shinkansens do not stop at some of the main, inner city stations such as Osaka. For your purposes, a hotel near Kyoto Station, Shin-Osaka Station (outside of Osaka) or Shin-Kobe Station would work best, since you will basically just be sleeping there. I can't think of an American brand near any of those but there is an ANA Crown Plaza next to Shin-Kobe Station that has reasonable rates. ANA hotels are pretty used to western visitors and Shin-Kobe is pretty convenient if you are using a Rail Pass.

Hotels like Hilton or other American brands will not be convenient to the shinkansen stations. I have stayed at that little business hotel near Shin-Osaka that I referenced up-thread so I used it as an example. It has no amenities other than being right across the street from the station and they will not speak much English. Maybe something else would work better for you.

The trip from KIX to either Kyoto or Shin-Osaka upon arrival is very straight forward, just a direct ride on the JR Haruka airport express. Shin-Kobe is almost as easy but would require a transfer at Shin-Osaka from the Haruka to a Hikari or Sakura shinkansen for the short run onward to Shin-Kobe. If you get the Kansai Area Wide Pass (not the similarly named Kansai Area Pass which is more restrictive) at the JR Service Center at KIX, the agent can reserve seats for the Haruka to Shin-Osaka or Kyoto or both the Haruka and the onward shinkansen to Shin-Kobe, all at once.

If you decide to cram Miyajima into your single day at Hiroshima, the Kansai Area Wide Pass will not work there. You would have to pay for the local train to Miyajima-guchi (the ferry port) and the ferry to Miyajima. Neither is very expensive.

For reference, the shinkansen to Hiroshima from Shin-Kobe will take about 1.25 to 1.5 hours each way. To Kyoto will be about 30 minutes if you can get a direct train, 45 minutes if you have to transfer at Shin-Osaka. The Haruka from Kyoto to KIX is about 75 minutes.