No, I lost you for a minute at first. Sauerkraut is high in probiotics as it's fermented. I spoke too soon and corrected what I meant when you asked how much.
If chickens actually eat this stuff on a regular basis, then I'd guess a tablespoon or so per day is plenty to give them the probiotic benefits. It's high in salt so that's why I'd not give them a huge dish of it. For my fermented grains, that's got way more nutrients in it, plus probiotics and prebiotics, and even with that, I don't let them gorge themselves as they need room to eat bugs and their own food.
For yours, for the idea of it helping to coat the screw, like
@LaFleche said and as I corrected myself, as much as you can get her to eat.
As to how to get her to eat it, I think I'd first drain it the best I could, then mix it in some Greek yogurt, or...
I'd mix up some eggs in a bowl, add the drained sauerkraut, then fry it, or...
(the least healthy) make some pancake batter, mix the drained saurkraut in it, and make some pancakes.