There are some modifications to the KL divergence that make it acquire some of the metric properties (though not all).
For example, the Jeffrey’s divergence modifies the KL divergence to make it symmetric.
There are some special cases see :
"Unfortunately, traditional measures based on the Kullback–Leibler (KL) divergence and the Bhattacharyya distance do not satisfy all metric axioms necessary for many algorithms. In this paper we propose a modification for the KL divergence and the Bhattacharyya distance, for multivariate Gaussian densities, that transforms the two measures into distance metrics."
 K. Abou-Moustafa and F. Ferrie, "A Note on Metric Properties for Some Divergence Measures: The Gaussian Case," JMLR: Workshop and Conference Proceedings 25:1–15, 2012.