Thanks for query
First of all from the history given by you it seems to me your daughter has asthma
but it should be confirmed by spirometry
with bronchodialator reversibility testing.
Secondly if it is proved to be asthma, it is not well controlled.
Look diagnosis of asthma is mainly clinical and clinically your daughter seems to have asthma. On the spirometry testing if obstructive pattern is seen with bronchodialator reversibility then your daughter's asthma will be confirmed.
To achieve control your treating physicians already stepped up the asthma therapy to step 4 as per recent GINA guidelines. Controlling asthma symptoms is the most important aim of treatment because virtually there is no cure for asthma.
Common triggering factors are house dust mite, cockroaches, dust, pollen, pollution, smoke etc. Keep your daughter aloof of them. You can consult allergist to detect the allergen by performing skin prick test and specific serum IgE assay and can combat them by giving specific immunotherapy