Magento Enterprise sometimes runs different patch versions than Magento community. For example, SUPEE-5344 is not available for your Magento version. I'm also not completely sure which checker service you are using, but they mainly focus on Magento Community and there is a possibility that a patch cannot be detected for various reasons. If the status response is "unknown" you should verify another way. If the status says "vulnerable" there is a big chance you still need to apply the patch. Refer to this quote from the Magento patch page:
If you are on Magento Enterprise Edition 18.104.22.168 and lower, then please apply SUPEE-1533 and the appropriate patch for your version:
You can download patches for Enterprise within your Magento account under a downloads tab. Verify that you have all those patches corresponding to your Magento version installed. I don't have access to this, but this repository has a nice list as a reference. Do download them from your account though to make sure you get the right ones. https://github.com/brentwpeterson/magento-patches/tree/master/EE1.12
Applied patches are logged in an
applied.patches.list file in the app/etc folder. The installed patches should be listed there. To be absolutely certain a patches is installed you could verify it manually. In the patch file itself there are lines indicating which file is being patched. The "+"'s and "-"'s indicate what lines of code are to be removed or added. Verify that the "-" lines are gone and the "+" lines are added. If your still in doubt, it's a good idea to get a certified Magento developer to take a look at it.