Not sure about the Tylan 50. I imagine it wouldn't hurt her as long as you don't overdose (kind of like Tylenol for humans). I'm not an expert on this, so you might want to search it in the forum for more information. As far as the disease, it sounds more like Coccidiosis to me than anything else. This is a very common disease for young chicks that is usually fatal if not treated quickly. Coccidiosis is caused by oosts that live in your soil and in the poop of adult birds, and chicks can get it very easily if they are transferred immediately outside from an indoor, dirtless environment. It is important for the chicks to develop immunity to the oosts when they are young, but going from no contact to full contact with outdoor soil can often cause the disease. Even if they have already developed some sort of immunity to the oosts, it is still possible for the chickens to get Coccidiosis. I had a fully grown chicken who lived outdoors all her life who just died yesterday, and it looks like Cocci. was the problem. Because your chicks do live outdoors, I am inclined to think that your baby has become infected with Coccidiosis as opposed to something like Mycoplasma.
What I would suggest is taking your chick to a vet (if you can afford one and have one near you) and find out for sure what it is. You will need to get medicine for the chick anyways, so going to the vet would allow you to get a proper diagnosis and get the drugs at the same time. In the meantime, separate her from the other chicks to prevent whatever it is she has from spreading. If you can, get her to eat plain, unsweetened yogurt (which will slow down Cocci if she has it and not do anything if she doesn't). I don't know of any home remedies for Mycoplasma, if that's what she has, so if you don't find anything that says not to I would give her a small dose of the Tylan 50 orally. For the other chicks, put about a teaspoon of Apple Cider Vinegar into their water. This is a good thing to do even if they are not infected as it will give their stomach some acidity for their "baby" years and allow them to fight off harmful bacteria until their bodies can do so on their own.
Do not delay the visit to the vet! Coccidiosis kills quickly, and even if this isn't what your chick has, other diseases should be treated as soon as possible to prevent further damage. Know, however, that even the vet may not be able to help you, and there is a chance your chick won't make it. If you don't feel like you can afford a vet visit, then do what you can to treat her on your own at home. Don't feel bad-- you're not being a bad chicken owner by doing this!
Best of luck! Let me know what happens!