The p-value is defined as the probability, under the assumption of hypothesis H, of obtaining a result equal to or more extreme than what was actually observed (Wikipedia). By "a result" it is intended a particular statistic.
For example, if I want to fit a distribution of data ($N$ items), I would choose the parameters that maximize the likelihood. Then, if I want to compute the p-value I can use the Kolmogorov-Smirnov distance: I compute it between the CDF of the empirical distribution and the CDF of the fit; then I sample datasets of $N$ items from the fit distribution, fit them, and compute the KS between the samples CDF and the CDF of the fits of the samples. Finally I estimate the probability that the KS of data-fit is bigger than the KS from samples, which would be the p-value.
Now, the KS is a distance between two distributions. Can I use the Kullback-Leibler divergence instead of the KS? In this case, it would be the Likelihood itself. So I would sample from the fit distribution, apply maximum likelihood and estimate the probability of the likelihood of the data w/r the model to be as extreme as the likelihood of sampled datasets w/r to their model.
What would be the problem with this approach? Is the $\chi^2$ analysis the same for gaussian-distributed data? Can I do the same for other distributions on the basis of the Wilks' Theorem?
If is not clear, please tell me, I can expand/explain.