I'm sorry you're having trouble.
Do you know what the shot was that the breeder gave the other hen?
Do you have any idea what the sick hen has (type of respiratory illness - Infectious Bronchitis, Mycoplasma, etc.)?
Antibiotics will only treat secondary infections that are due to complications of respiratory illness, they usually are not given as a preventative. Usually the method of treatment would be to only treat chickens that are showing symptoms of illness.
To answer your question, yes, mix your Tetracycline and leave it for her to drink free choice. This needs to be the only source of water available during the treatment period.
FWIW depending on what type of respiratory illness your hen has, it can make her a carrier of the illness for life. The hen that is showing no signs of illness at this time, if housed with the sick one, would be considered a carrier as well - even if she never shows any symptoms.
Keep us posted.
http://edis.ifas.ufl.edu/ps044