It looks like hot springs are a high priority for you, so there really are tons of places across the country. The closest place to Fukuoka is in Futsukaichi, which is just 15 minutes by train from Tenjin Station, and has a history going back 1300 years. It's a far cry from Kurokawa or Yufuin, but a nice place to spend an afternoon if you want more. It's also very close to Dazaifu, which many visit for its Tenmangu shrine and Asia museum.
There's even a few in the city, or Seiryu Onsen is Nakagawa, a neighboring town.
http://www.nakagawaseiryu.jp/Given a choice, I think I'd pick Kurokawa over Yufuin, for its better forest ambiance and outdoor baths. If you go to Yufuin, you could go to Beppu as well though, and there is more to see there besides the hot springs, like Hells Tour, and nearby Takasakiyama monkey park/aquarium.
http://bit.ly/2pZ9opzYou also might consider the Unzen hot springs in Nagasaki as well. Plus Nagasaki City is really the most interesting city on the island, and since you are flying all the way into Kyushu, it's worth considering going there. Fukuoka has its sights as well, especially the Nanzoin temple which is one of Japan's best yet mostly unknown to the outside world.
http://www.thejapanfaq.com/fukpic.htmlhttp://bit.ly/1KAm8cUThe highlights of Miyajima and Hiroshima can be done in a day. You might like an overnight at Miyajima - once all the day trippers leave, the whole area becomes very quiet and tranquil. Also be sure to go to Itsukushima Shrine both at high and low tides. The shrine is best worth visiting at high tide, and at low tide you can walk out on the beach right up to the giant torii gate.
http://www.tides4fishing.com/jp/hiroshima/itsukushimaIf you do, you also might have time to see Iwakuni just before Miyajima. It is quite delightful.
For Koyasan though, expect a lot colder weather (snow) and the JR Pass won't help you - get the Koyasan World Heritage Ticket.
https://www.japan-guide.com/e/e2358_012.htmlIf you go to Takayama, it might make more sense to see Kanazawa first. You can also see Shirakawago along the way, although you need to pay for the bus.