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Doxycycline can be used in cats and is often used to treat FIA and chlamydia. You would need to check with a pharmacist or your vet as to dosage. Dosage is based on weight, not age. Note that reconstituted (liquid) Dox is only good for about 2 weeks after mixing, so if this is something left over from an earlier treatment, it will not be effective if it's been mixed for over 3 weeks.
Hopefully this answers your question. For detailed specifications, a vet or pharmacist must make that decision. Also, this is probably not the drug of choice for a bite wound. There are others that would be more appropriate and effective.