Are you using Chrome on your desktop to automatically reload the page on save? If so, I'm surprised that workflow isn't working currently for you. Can you point us to more information about the live reloading feature of CodeKit?
You might try checking out this blog post on this topic: http://blogs.adobe.com/edgeinspect/2012/04/24/shadow-livereload-workflow/
I am using chrome. I believe the difference between using codekit and LiveReload, is that LiveReload has the functionality to reload all the connected devices when saved, whereas codekit will only reload the browser on the main device(desktop for me) when saved and that to reload the connected devices (iPad, iPhone, iPod) I have to manually reload the browser on the main device.
Here's the link to the page about CodeKit's live-browser reloading feature. http://incident57.com/codekit/help.php
I was using LiveReload previously in the same way that the blog post you inserted talked about. But because Codekit offers a few more features when it comes to preprocessors and image compression, I was hoping to drop live reload and use codekit exclusively. Unfortunately, like I stated above, the one problem I'm finding with codekit is that when using adobe edge inspect, is that codekit won't live reload all the devices when i save in the code editor, it will only live reload the browser on the main device, whereas LiveReload will update both the browser on the main device as well as all the devices connected through edge inspect. With codekit, I have to click on the browser on the main device(desktop) and manually reload it to update all the external devices.
I may just use both, LiveReload for the automatic updating on all devices, and then codekit for all the other features.