Perfect! The directions it comes with say to mix it in the water, but Baytril is quite bitter and most birds won't drink the medicated water, so I strongly suggest that you give it orally instead.
Can you buy some Baytril? That is probably the best drug for treating this. It is banned for use in poultry, but many vets still prescribe it, though you can get it without a prescription quite easily.
https://www.backyardchickens.com/articles/baytril-enrofloxacin-sources.74374/
The amoxicillin would be better. If you do decide to give the Tylan, it too must be given more than once a day. Most vets recommend 3-4 times a day for that, and the dose is no less than 0.2 ml per pound of body weight 3-4 times a day, so a five pound chicken would get 1 ml 3-4 times a day, not...
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Some vets will, some won't.
It's tricky... Baytril is actually banned for use in poultry, but some vets still prescribe it. If you find a vet that won't prescribe it there are other drugs that might work as well. Such drugs might be Amikacin or SMZ/TMP, or a combination of drugs.
Tylan will not treat E. coli or pseudomonas. Tylan is for treating gram positive bacteria and if she has an infection, it is almost certainly a gram negative infection, for which something like Baytril is used.