What are some well-known asylums in Japan?

You might be writing a story like "One Flew Over the Cuckoo's Nest "?

But if it's a fiction, it would not be necessarily a real name.

You should remember that using real name would take some 'risk', because your scenario would include a very delicate issue, which is totally different taste from, say, those using 'Alcatraz', 'Guantanamo' etc...

You need to be specific what kind of "asylum" you want to know.

I don't think there is an asylum in a place "no one can find".

If there are few people in one place, there would be no asylum there.