These replies seem to be getting gloomier and gloomier...
I have only been to Kagoshima a few times, and never to Miyazaki, and I don't have any particularly brilliant suggestions, but if you already have plans to visit this part of Japan I wouldn't just cancel them. For one thing, it might NOT rain!
All of my visits to Kagoshima have been in June, for a total of around 8 days, and it only rained on a couple of those days, and in both cases not all day. The day it rained hardest I had plans to go to Healthy Land Spa in Ibusuki via the Ibusuki no Tamatebako tourist train. I decided to go despite the gloomy forecast, and it was definitely worthwhile, although I have to say it would have been better in a lighter rain (it poured buckets most of the time I was there). The train was a lot of fun anyway, and the rotenburo was phenomenal. (They also have a sand bath but I didn't try it.) The main difficulty I had was walking from Healthy Land to the bus stop during a period when it was really pouring, but even that ended up being okay (I had a fairly good umbrella, and it served me very well). Anyway, it was one of the best outdoor soaks I have ever had. (I even had the bath to myself for part of the time.) So I would say this is something you could do, possibly in conjunction with other things in that area. You really have to do your research regarding transit (in particular buses, but they are actually pretty workable), and if you want to take the Ibusuki no Tamatebako you should get seat reservations in advance. (Worst case you could take a local train, but they stop a lot and aren't as cute.) Also read the Healthy Land pages in detail, noting closure days for their various facilities. https://www.seika-spc.co.jp/healthy/en/
For a rainy day that is not too stormy (as in high winds and excessive amounts of rain, which could possibly disrupt train transit and leave you stranded for many hours someplace in the middle of nowhere), consider recreational train riding. The JR Kyushu Special Trains that run between Kagoshima Chuo and Kumamoto (Hayato no Kaze, Isaburo-Shinpei, Kawasemi/Yamasemi) can make for a memorable day, and although the visibility will be diminished by clouds and mist, the scenery is still stunning, the trains are very cool, there are some very neat old train stations, and there are some nice bentos for sale along the way. I have done this route twice (once in each direction) and highly recommend it. But you really have to do your research as far as schedules go. And reserve seats. (Of course, if you are leaving this as a fallback only for a rainy day, it might not be possible to get them at the last minute.) The JR Kyushu Special Trains page is a starting point: https://www.jrkyushu.co.jp/english/train/train_index.html
If you are in Kagoshima, maybe you could go to the Tourist Information Center in the Kagoshima Chuo train station and see if they will help you plan this trip. (They can also suggest other rainy day activities.)
Two other possibilities for Kagoshima City are the aquarium and the Tenmonkan covered shopping arcade. The arcade is nothing terribly special but you can poke around there for a couple of hours and have a meal there, and it is protected from the rain, at least.
Of course, the train riding suggestions will be more appealing if you have a rail pass. Trains can get pretty expensive otherwise. But as far as I'm concerned, the fact that "rain happens" is one of the reasons why it's worth buying a rail pass even if you "do the math" and come up with individual tickets for your planned travel being a little bit cheaper than a pass.