If you are seeing egg contents, it sounds like she is expelling naturally. I would not be inclined to internal examine her at this stage, you will only cause distress. Internal exams should only be undertaken if you have a understanding of the physiology and anatomy of a hen as well as a clear understanding of the egg laying process. Saying all that, a gentle external feel, and visual observation can tell you quite a lot.
Most commonly in a young pullet it is the soft egg, or incomplete egg that causes problems, leading to discomfort, egg binding and peritonitis.
The best you do can do right now is ensure she is hydrated, so plenty of fresh clean water. If you can get hold of limestone powder, an easily absorbed and quick form of calcium this will not only help with egg shell quality but also with the muscles which are involved with egg laying. You could give her a warm bath to ease her muscles but to be honest I imagine in this weather she will be warm enough!
What to look out for... inappetence, pallor, abdomen feeling warm, fluffed up, generally her condition not improving or worsening. This could indicate an infection,or certainly a further problem which will need diagnosis and treatment from a vet.