Where do people go after leaving Eikaiwa?

From what I've heard everybody who works for one of the big schools (Aeon, Shane, ECC etc...) leaves for a better job after 1-2 years.

Where do these people usually go?

I'm no expert, but I don't think there's any "usual" place that people go to.

It's just that, for people coming from English-speaking countries with no connections, being an eikaiwa teacher is one of the easiest ways to start his/her living here in Japan.

But maybe they didn't want to be eikaiwa teachers. Maybe they wanted to be translators, or lawyers, or performers, or housewives/househusbands. But they can't just come here without a job/sponser, so they do eikawa first.

And while doing eikaiwa, they gradually adjust to the culture and learn the language and make connections, and sooner or later move on to jobs they like better, or jobs that pay more, or jobs they feel that are more rewarding.

That said, there is a certain percentage of people who love working as eikaiwa teachers more than anything else, and keep the job forever in one way or another. Some start their own eikaiwa business.

I think you misunderstood.
English teachers usually go back to their home countries after couples years and they find a better job over there.
A typical English teacher with a degree in East Asian Studies won't find a better job here, usually.

Mental hospital....

From my experience the biggest factor is Japanese ability.

Those who speak Japanese usually go into translation, finance, head hunting, IT, real estate, etc.

Those who don't usually go work for non-eikaiwa schools (university, high school, etc.) or go home.

There aren't many options when you don't speak the national language.