In the picture of the lash material, the top looks like lash egg, the bottom may be a malformed egg shell. Lash material is from salpingitis, which is infection/inflammation of the oviduct. As sick as she looks, it's probably advanced, and likely will not respond to medications, but you can try. Sometimes it may buy them some time even if it' doesn't cure it. It's notoriously resistant to treatment. If you want to try to treat then enrofloxacin is often what is recommended, you can get that here:
https://jedds.com/products/enrofloxacin-10?_pos=1&_sid=67a6fd1d7&_ss=r
But I would also prepare yourself that you may lose her. The leaking vent is likely a symptom of what is going on inside, it can make it hard for them to push droppings out so they get messy vents. If there are flies active where you are then flystrike is a risk, from flies laying eggs in the droppings stuck there, so if you're going to try to treat her, then you need to clean that off and keep it as clean as you can. I would not bathe her, she appears too sick and that could put her over the edge. Just use warm, wet cloths and dry her off when done. She may not want to eat as it can also slow their digestion and crop problems can accompany it also. That's also caused by what's going on in the abdomen.
More on salpingitis here:
https://the-chicken-chick.com/salpingitis-lash-eggs-in-backyard/