Located in Kirishima near Myoken onsen, Wake-jinja and Inukai falls are easy to visit if you are staying for the night at a ryokan near the onsen.
Both are near the start of the former Kirishima Olle trail, but talking with the staff at my ryokan, I was advised to not do the hike at the risk of getting lost in the forest. While I walked from the onsen to the shrine and waterfall, I did made a small navigation mistake and ended up on a part of the old trail leading to the waterfall and not on the main road. While I knew it would get me where I wanted to go, there was several trees in the trail that you have to climb over or crouch under, so I would not recommend, especially as I figure it can only get worst unless someone decide to revitalize the hike or if you can get a more up to date report on the status of the trail. Of course, if you go by car you will have to get there by the main road
On the other hand, the path and staircase from the road to he waterfall are well maintained, so I would for sure take time to go see it from closer. Both the waterfall and the shrine have their own parking, but they are close from each other.
The shrine have nothing spectacular and likely best visited in late mid to late April to see he wisteria. I was there April 20 and it just started to bloom, one more week and it would probably have been a better timing.
Appart from the flowers, there is no strong reason to visit, unless you are a Sakamoto Ryoma fan, as his was part of his honeymoon with his wife, Oryo, and the shrine have a shuin with the shadow of the couple on it.