I've enabled the plugin and it's working fine right now when I add it to the sidebar: https://i.imgur.com/sCP5Qm2.png
Your purchase code shows me a purchase date and time Sun Dec 07 22:00:13 +1100 2014 which is not corresponding with the search version you use. Your site has plugin version 3.1 installed, which should not be available from 7th of october. On 7th of december you should have gotten version 3.5.
In case you are using a 3rd party pirated version, please completely delete it, it's very dangerous. 3.1 is a well known pirated copy and most of the free pirated downloads can contain malicious code, which steals your server credentials, like mysql passwords, admin passwords, corrupts the file system and installs backdoors to your wordpress installs. If that's not the case, then my apologies.
Either way, please upgrade to version 3.5 which should be available on codecanyon if you download the plugin. An bugfixed version was uploaded just yesterday. The documentation contains information about how to upgrade, which you can find here: http://wp-dreams.com/demo/wp-ajax-search-pro3/docs/#important_upgrade_notes
Let me know if you still experience the issues after the upgrade.Best,
If you like my products, don't forget to rate them on codecanyon :)