language_attributes for two languages?

You can use a filter to modify the value before it is set

function __language_attributes($lang){

  // ignore the supplied argument
  $langs = array( 'en-US', 'KO', 'JA' );

  // change to whatever you want
  $my_language = $langs[0];

  // return the new attribute
  return 'lang="'.$my_language.'"';
}

add_filter('language_attributes', '__language_attributes');

Then just make sure your theme header has the correct php function

<!DOCTYPE html>
<html <?php language_attributes(); ?> class="no-js">
<head>

Another approach would be to set the language in a global variable before the filter is called.

// Make this variable global
global $__language_attribute;

// Set the language we want to use
$__language_attribute="en-US"; 

// Listen for the language filter
add_filter('language_attributes', '__language_attributes_use_global');

function __language_attributes_use_global($lang){
   global $__language_attribute;
   return "lang=\"$__language_attribute\"";
}

ALTERNATE

// set your language here
$my_lang = 'KO';

// subscribe with closure to apply this value later
add_filter('language_attributes', function($lang) use ($my_lang) {
    return 'lang="' . $my_lang . '"'; 
});