Rather than Woolite, which can actually be quite harsh on wool try a small amount of shampoo or shower gel in just slightly warm water.
lay the kilt as flat as possible and gently pat the water through it, paying particular attention to where there are the most layers.

If you can lift the kilt out to let out the water and refill that would be good, as it minimises the agitation of the fabric. I have a folding drying rack which can be laid flat across the bath. Check that the water is at the same temperature as for washing then rinse gently twice, then again adding a cup of white vinegar to the water. Wool likes to be slightly acidic and the detergent is alkaline so the vinegar will counteract that.