Yes, the twisted up parts, to me, appear to be the membrane of the egg that broke and got twisted up - possibly before she was able to expel it. It could also be that when she lays, the egg is so soft that it breaks on impact and you’re left with the twisted up looking membrane. One of my chickens has had this happen in the past, she would also finally get the soft eggs laid from the roost at night, as yours does. This is somewhat like what ours would lay, see the membrane part?
Layer feed is good, but if you are feeding anything besides layer feed, such as scratch or meal worms, I would stop. We’ve had great luck getting our hen to go back to normal laying when we cold turkey stopped treats. It could also be shell gland issues or infections causing that, but it’s hard to say.
It sounds like from what you’re describing, she may be egg bound and struggle to push these soft eggs out. If she were mine, I’d give 600mg of calcium citrate plus d3 anytime she looked in pain and seemed to struggle with laying. You can get them at almost any grocery or drug store.
Just gently pull her wattles down and pop a pill in her beak. These will help stimulate contractions and supply some extra calcium. I usually give once a day for a few days after an egg bound or soft shelled incident but no longer than a week. As for her limping, I’ve read where it’s possible for stuck eggs to push on leg nerves and hurt. I know my hen always holds one of her legs up like a flamingo when she becomes egg bound. Does the lameness resolve after she lays?
Edited to add, also very important: if you don’t already, always offer oyster shell in a dish available for your chickens.