I don't think the frayed tail feathers will tell us much; that could easily happen if she's turning around & around in the nest box. As far as the droopiness and allowing herself to be picked up with no fuss whatsoever...that's not good. I definitley think you have an illness going on. What, I'm not sure. It seems there are SO very many poultry illness that present with lethargy, like this. If she were mine, as a first line of defense, I'd go ahead & give her a treatment of Ivermectin Eprinex. Eprinex is the VERY BEST Ivermectin available - it's expensive, but well worth it. I tried to be cheap & get the regular topical one that was less expensive, and ended up with mites again in short order. The Eprinex will rid her of any unseen parasites, as this could be a contributor.
Again, if she were mine, I would first treat with Eprinex and then move on from there - reason being, one of the most common ailments is parasites, which can cause issues with egg laying.