Thanks for adding those IPs. I checked out the site and can see that the problem is a lot of JavaScript errors being thrown in the console. Some area coming from plugins, others from your theme. It is highly likely that one of these conflicts (probably from your custom theme) is responsible for breaking the AJAX/jQuery based search function in the WP Knowledgebase plugin.
To confirm this, follow standard steps to isolate the error. That means switching to the default twentytwelve theme and deactivating all your plugins except WP Knowledgebase. Test. Does it work now? If yes, then that confirms it was a conflict in one of the items now deactivated. You then switch things back on, one by one, testing in between, until you find the culprit.
Once you have identified the source of the error you can troubleshoot further to fix those conflicting scripts.