Hey @b-lynk,
Sorry for the lack of info. I thought you already knew how to setup a Navigation Inline element because you already have set up one in your header.

I recommended that you try the Navigation Inline element because WPML says that the Language Switcher will work with menus that supports drop-downs. The menu placed in a widget does not support a drop-down.

What I was saying is you just need to assign the language switcher menu in the Navigation Inline Setup. Please check the screenshot below taken from my previous screenshot.

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I have executed the steps in your site and you now have a Language Switcher. Before that, I experienced a problem though. Details below.

I thought there is no issue with the language switcher setup until I logged in to your site and found that an empty menu will not show up in the Assign Menu dropdown in the Navigation Inline > Setup so what I did is I added 1 custom link in the language switcher menu and added the code below in the Navigation Inline’s Element CSS as a temporary workaround.
$el > .menu-item:first-child {
display: none;
}
Please note that I do test in my test site first to ensure the issue is not a bug in our theme.
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We’re sorry for the back and forth here. We don’t intend to do that or waste your time.
Hope that helps.