I have a large sample size of about 50,000

I want to determine what theoretical distribution fits the distribution of my samples the best. What I did is to fit all distributions I know to the data and check which one has the lowest SSE. I'm personally quite happy with this as the distributions with low SSE also seem to fit the data graphically quite well.

Anyhow, I've realized that most researchers use a statistical test to "proof" that one distribution fits better than another. I thought maybe I could use the KS-Test on the best distributions and compare the p-values. On the other hand the sample size is quite large and I've heart in cases like this effect size might be more interesting. So I thought of comparing the different distributions using the Kullback–Leibler divergence.

What do you think?