The last RI issue I had, I was given 1mL Tuberculin syringes with 25 gauge (5/8" length) needles. The size of the syringe you need will depend on the dosage of Baytril. I had a different antibiotic, so I can't specifically help there, but I'd be surprised if a dose was larger than 1mL (my girl was over 700g and her dose war 0.3mL). I definitely don't see any reason to use larger than a 25g needle, though (smaller number = bigger diameter needle).
I do second the need to get a culture done, though. Your options are obviously very limited, and Baytril could fix it, but I'd get a culture swab and then dose with Baytril until you get the results that tell you whether to switch treatment, if at all possible.