How can I restore Korean language on my iMac keyboard?

A couple of days ago, my desktop logged me out of several sites. When I log back in, it acts like I was using this computer for the first time, but I've been using my desktop for years. And then, I noticed that the Korean language option is no longer showing in my menu bar.

I tried adding from Input Sources, but it only shows U.S. I click the plus button, but Korean is not there to add

English and Korean are listed in the Language & Region section, but not among the keyboard input sources. Three-quarters of my business is in Korean, please help!

[Edited by Moderator]

Original Title: Keyboard language changes unexpectedly