You most likely will need to edit your theme files to do that and use the php shortcode to display the search.
It's usually the header.php file in the theme folder, but it varies from theme to theme. Before editing any of the theme files please make a backup in case anything goes wrong.
Based on the html output I can see, the search bar should probably be placed between the navigation menu and the header links div inside the “header-menu” div into a new container displayed as a table cell, something like:
<div style='display: table-cell; vertical-align: middle'> <?php echo do_shortcode('[wpdreams_ajaxsearchpro id=1]'); ?> </div>
But this is just a guess based on a quick look at the theme. It might be much more complicated than this. If you want to be sure to have a correct implementation, you can always ask the theme developer if he has a better suggestion.Best,
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