The normal distribution is sometimes (maybe even often) a reasonable simplification of reality, but it is almost never going to be true in a strict sense. So, your Jarque-Bera test being significant did not tell you anything you did not already know before you perfomed that test.
The real question you should ask yourself is whether or not the deviations are large enough such that the normal distribution is no longer a reasonable simplification of reality, and statistical tests don't answer that question. Instead you should just look at the residuals (I like QQ-plots for that) and make your own subjective judgement. You should then document how you came to that decision, so that others can decide for themselves if they would have made the same judgement call as you did. This being able to trace back how you came to a decision is what makes your work "scientific".