There are a lot of guides out there to show you how to add language packs to Windows, but I've found it can be sometimes annoying to try to set the languages right. In my experience this is the easiest and most full-proof way to get it right. So here we go.
Go ahead and click these screenshot thumbnails for full-size views.
Go to Windows 10's "Time and Language" settings.
If you haven't done so already, click that big "+" sign to Add a Language, and add Korean.
That's it, you don't need to mess with any of the settings in there. It will probably say it's contacting Windows Update to find the language pack. Cool. Give it a minute to do that.
Then click "Additional date, time, & regional settings,*" highlighted above.
Click "Change input methods".
Yeah you could have gotten here several other ways, but this is the way I like to do it.
Click "Advanced Settings" over on the left-hand pane.
Here's where the magic happens. Set these two menus, the first to English, the second to Korean.
Enjoy typing in Korean.
I like this method because it forces Windows to override whatever wacky language settings you might have and forces it to use the Microsoft Korean IME for inputting Korean.
Don't worry, that doesn't mean you can only type in Korean. The Korean IME is the tool you need for easily using that 한/영 key on your keyboard, so it lets you flawlessly switch between the two. No more having to click to menu and change the input method with your mouse every time you start-up your computer. Seriously, I saw my friend do that. This little guide is dedicated to him.
*Props to Microsoft for using the Oxford comma before an ampersand. Bold move.