Thanks so much for your help. It is definitely getting closer to what we are looking for. I have implemented your code here:
Is there a way to have the search box appear below the icon like it does in the default theme search. This is visible here on our staging area site : http://mather.staging.wpengine.com/
Also we would like to use the same icon (the white magnifying glass) which is used in the staging site link above to trigger the search. Ideally the search would look and feel similar to the standard theme search but perform with your search's awesome functionality.
Lastly, when the browser window is made narrower, there is an issue with the placement of the search icon. Do you have any suggestions for this?