Is there any way to measure of Gaussian-ness?
One way to measure the similarity between two distributions is the Kullbeck-Leibler divergence. Granted, it is not a norm because it is not symmetric, but it is a way of quantifying the distance between two pdfs. 

EDIT: There is a topic called "Normality testing" or "Gaussianity testing" that is devoted to that very task. Perhaps some tools from that field will help.