Dried mint leaves and dried mandarin peel made into a tea with honey three times a day should relieve the fluid congestion and ease the cough or at least make it more productive and less irritating.
Mint should not be consumed for very long periods of time more than a few days in large amounts and it certainly shouldn't be used at all if you have any of the following - night sweating, hot flushes (non flu associated), dark and scanty urine (again, usually and not while you have had the flu/bronchitis), insomnia and general dryness.
Dried Mandarin peel is as safe as houses to have daily in quantities that are quite large without problems.
This will not resolve your asthma but merely assist your body to break up the phlegm and reduce the heat in your body (evidenced by the colored sputum). You should take the full course of antibiotics.
Massage the following points three times a day for a minute each on both sides of the body:
San Jiao 5 (WeiGuan TE 5 on the picture) to expel exterior pathogens:
http://www.jan.000a.biz/health/wuweipin ... 99F20.html
Stomach 40 (Feng Long) to treat phlegm congestion:
http://www.acatcm.com/stomach-meridian- ... -feng-long
Ren 17 (ShanZhong - scroll down until you get to it) to open and move qi in the chest to relieve congestion:
https://quizlet.com/45823274/ren-meridi ... ash-cards/
Large Intenstine 11 (QuChi) to clear heat:
https://www.pinterest.com/pin/548524429586926361/
If you want a more detailed examination of your asthma, I can help you with that once the bronchitis has cleared up if you like.
Also, I have found heart chakra meditation helps when the lungs are congested especially if it is a contagious viral or bacterial cause.