Hi Jon,

I understand your issue and we have not implemented that functionality, since customers haven't requested. However we are in the process of implementing this functionality, our designer has designed this page already:

(view large version)

To add the search functionality in the menu (hard fix) please add the following code:

<style>
    .search-input-menu {
        float: right;
        margin-right: -15px;
        margin-left: 25px;
        z-index: 10000;
        position: relative;
        pointer-events: all;
    }
    
    .search-input-menu .search-input {
        position: absolute;
        right: -10px;
        top: -8px;
        width: 200px;
        visibility: hidden;
        opacity: 0;
        transform: scale(0);
        transform-origin: right center;
    }
    
    .input-visible.search-input-menu .search-input {
        opacity: 1;
        visibility: visible;
        transform: scale(1);
    }
    
    .search-input-menu .search-input input[type="submit"] {
        display: none;
    }
    
    .search-input-menu .open-search {
        font-size: 18px;
        color: #333;
    }
    
    .search-input-menu .open-search:after {
        display: none;
    }
</style>
<script type="text/javascript">
    jQuery( document ).ready( function( $ ) {
        $( ".search-input-menu a" ).on( 'click', function( ev ) {
            ev.preventDefault();
            var $parent = $( this ).parent();
            
            $parent.toggleClass( 'input-visible' );
            setTimeout( function() {
                $parent.find( '.form-control' ).focus();
            }, 200 );
        } );
    } );
</script>
<div class="search-input-menu">
    
    <div class="search-input transall">
        <?php get_search_form( true ); ?>
    </div>
    
    <a href="#" class="open-search">
        <i class="fa fa-search"></i>
    </a>
</div>

And then add this code in: wp-content/themes/kalium/tpls/header-main.php on line 283 after "menu-bar" item:

(view large version)

This is what I can offer for you, for more you must utilize this code.