Implement Search Live in the menu and be shown when clicking a link

I am working to include Search Live in the menu of my website. My idea is that when you press a magnifying glass button, the search engine is shown to javascript.

The problem is that by hiding the div with display: none; if I make the call to the menu with echo do_shortcode (‘[search_live]’); It always shows me the menu, pressing or not the button.
If in the div display: none I put html code if I hide it and it appears, what can I do?

Thank you.