First get the terminology right: you are considering Likert-type items (not Likert scales which are sum scales of Likert-type items).

My first test would be chi-square of conditions crossed with Likert-type items; if nonsignificant, you're done. But if you find a significant result, so you reject independence, you need to address what is different between conditions: measures of central tendency (mode, median, mean) and/or measures of dispersion (range, variance). Typically, researchers are interested in means, and the associated tests such t or F. The problem of these test is not normality, as these statistics been shown to be fairly robust against deviations of normality in numerous studies. The issue is that if you find a significant difference of say, 0.3141527 between groups, what that means in real terms. The median test will give you the medians of the groups and the assessment whether observed differences are significant - and since the medians are expressed in term of the used scale they have an unambiguous interpretation.