In the Kullback-Leibler divergence, m (x) must be a probability density, whereas in Jaynes' formula, m (x) is simply a density, meaning that it does not have to integrate to 1.
In the Kullback-Leibler divergence, the density function m (x) is replaced by a second probability density function q (x). In Jaynes' formula, m (x) is not a probability density but simply a density, meaning that it does not have to integrate to 1.