I'm going on a trip to Kumamoto with my girlfriend. Can you recommend a high-class Ryokan (Japanese inn) we can take a hot spring bath and have lunch included?

I'm planning a trip to Kumamoto with my girlfriend. We've decided to go to Kumamoto Castle, but we also want to take a dip in hot spring somewhere. Kurokawa Onsen is famous, but it doesn't have to be Kurokawa Onsen. Can you tell me a high-end hot spring inn Kumamoto Prefecture where we can take a dip in hot spring with lunch included? Our budget is 20,000 yen per person. We'll go by car.

I want to explore the castle town and take a dip in the hot springs! Can you recommend a place to stay near Inuyama Castle?

I want to go to the castle town of Inuyama Castle in Inuyama City, Aichi Prefecture, which is a hot topic right now, take photos of cute sweets that look great instagrammable, and eat them! Are there any recommendations for accommodations within walking distance of Inuyama Castle?

After sightseeing at Inuyama Castle, I would like to stay at a nearby hot spring inn.

I'm going to Inuyama Castle because I enjoy visiting castles. I'd like to stay in the surrounding area, so please introduce me to some hot spring inns. If they have an open-air bath, that would be great.