I've run a Kruskal Wallis test, and for some of the questions the p value is not significant. Would I report this in the same way as if it was significant, stating the df, test statistic and p-value? So it would be something like this a Kruskal Wallis test was conducted but the results were found not to be significant H(3) = 2.119, p>0.05 (or would I state the exact p value here (.548))
Yes, non-significant results are just as important as significant ones. If you are reporting any result, always include the df, test statistic, and p value. And in that case, you should state the exact p-value, rather than generalising to >0.05
If you are publishing a paper in the open literature, you should definitely report statistically insignificant results the same way you report statistical significant results. Otherwise you contribute to underreporting bias.