Significant Result in Levene's Test I am very confused right now. I ran Levene's test on my data and got a p-value of 0.000, meaning that variances are very heterogeneous. I transformed the data but no method can make them homogeneous. So that's it, ANOVA would be inappropriate to use. I was thinking that my data could be nonparametric so Kruskal-Wallis would be the best test to use. However, when I tried testing the data for Levene's test for nonparametric data, I still got a significant result.
In short, my data failed the Levene's test for both parametric and non-parametric assumption. What do I do now?
 A: None of this follows ineluctably from the evidence you give.  

I ran Levene's test on my data and got a p-value of 0.000, meaning
  that variances are very heterogeneous.

Possibly so, possibly not. The result is highly significant, but that may just mean that you have a large enough sample size to allow firm rejection of the null. It could be that the difference in variances is not fatal to ANOVA. 
 I transformed the data but no method can make them homogeneous.

Possibly so, possibly not. We can't tell without looking at the data and hearing what you tried. Perhaps you missed out a transformation that would help. (I've seen people try transformations that make their problem worse; that need not be you, but you don't give enough detail for us to be sure.) 

So that's it, ANOVA would be inappropriate to use. I was thinking that
  my data could be nonparametric so Kruskal-Wallis would be the best
  test to use. However, when I tried testing the data for Levene's test
  for nonparametric data, I still got a significant result.

Data are not parametric or non-parametric, just techniques. That's a misuse of terminology. See notably @Glen_b's answer here More crucially, I don't know what Levene's test for nonparametric data means. What makes you think that Kruskal-Wallis requires any such prior test? 
I'd recommending that you back up and show us your data, or at least informative graphs, and tell us what interests you about them. 
