You've just managed to confirm the diagnosis of salpingitis. That gross "thing" is hardened pus from an infection in her egg "works". Now we know that penicillin is the way to go.
Be warned, though, even if she manages to recover from the infection, her egg-laying days are likely over.
I agree that it is salpingitis but in my opinion her oviduct is already impacted with egg and pus material (lash egg) like that piece you got out, which is why you see the swelling around her vent. That mass (and it will most likely be somewhere near the size of an orange but lumpy) is blocking her oviduct and constricting her gut and preventing her from eliminating waste properly, which is probably why she is reluctant to eat ie. she's bunged up. Even if she improves enough to be able to eat and expel waste properly, she will start to ovulate again at some point(release yolks from the ovary) and since the oviduct is blocked, they will just end up in her abdominal cavity (internal laying), where they will cause further swelling and eventually kill her. By all means try an antibiotic, but you may need to inject it rather than give orally if her digestive system is bunged up. Also, be prepared to euthanize if you don't see any improvement in the next few days because this will be causing her some discomfort. I have seen several videos of necropsies where birds were so packed full of this stuff, it really brought it home to you seeing it, how useless the situation was and how much discomfort the bird must have been suffering. I can find some and post links if you are interested and it may help you make a decision but they are graphic and not for the faint hearted.
I'm so sorry I can't be more encouraging about this. Salpingitis can supposedly respond to antibiotics if it is caught early. The problem is that by the time you see swelling and they are not pooping properly, it is generally too late.
Sadly chickens are all too often prone to these ailments due to the intense pressure their reproductive system is under..... Our modern hens have been selectively bred to far exceed the egg production of their natural ancestors and it is my belief that this is part of the price that we/they pay for that increased productivity.