As the phlegm is white then this is unlikely to be a bacterial infection and I would not use the Curam owing to risks of developing gastric side effects and resistant bacteria.
It is highly doubtful that this is Kennel Cough. Most adults pick up viral respiratory tract infections from the atmosphere and in a young healthy adult then this should improve over next couple of weeks.
Continue with steam three times daily, drink plenty of water and consider a decongestant if you still have nasal symptoms. Otherwise it will gradually improve over next two weeks.
Do you normally suffer from asthma?
What other medication have you been prescribed for your breathing?
Flixotide is a combination of steroid with long acting bronchodilator. You can check with your doctor whether you need to increase this dose short term while you have this infection.
It is not unreasonable to be also taking a short acting bronchodilator up to four times daily (salbuterol/albuterol/terbutaline) to help open the airways and reduce this sensation of reduced air exchange.
You should see a respiratory physiotherapist for techniques to bring up the phlegm without further irritating the airways.
As an asthma patient then you will have increased tendency to mucus production. Do not use cough suppressants or antihistamines as this will only make matters worse.