This sounds like classic asthma, esp in a child. Lots of children do not wheeze but cough is their prevailing symptom.
Night time cough and exercise induced cough also absolutely classic. Is thete asthma, eczema or hayfever in the family?
The blue inhaler is what's called the 'rescue' one, it opens up the inflamed, constricted airways to produce relief, but it doesn't get rid of the underlying cause, which is inflammation of the airways.
It is often prescribed by Gps to see if it makes a difference to the symptoms, but often it doesn't, and it only lasts 4 hours. It can be helpful to give before exercise or in the midst of a coughing fit, 2-3 puffs via a spacer ( the spacer is really important, otherwise the medication does not get delivered properly to the airways and that is THE most common reason people don't see a response.
The brown inhaler ( or "preventer") is a mild steroid ( it is so mild and only delivered to the airways, virtually none is systemically absorbed).
This will treat the underlying inflammation.
Now, the important thing to remember here, is that it will take about 3-4 weeks, often to see an effect. This is because the dose is low and is not systemically absorbed.
You HAVE to take it regularly, 1 or 2 puffs ( depending on what's prescribed) twice daily, via a spacer ( remember, the spacer is vital).
A puff here or there is utterly useless when we're talking about the brown inhaler. It is dependent on regular dosing to the affected area to reduce the inflammation. ( it's like only applying a cream to an inflamed area once or just when you remember, it won't work).
Asthma can ebb and flow over a lifetime, sometimes children "grow out of ir", only for it to be triggered later in life by a cold or something.
Don't fret about the montelukast, it is first line treatment for asthma in children, no medication is without any risk at all, but during trials and on the data sheet you get, they HAVE to report all symptoms while the people involved in the trial were taking the drug whether they are as a direct result of taking it or not.
It has been used for 2 decades or more and is very safe.
Hope this helps.
It's horrible to see your little ones struggling but hopefully, if you understand the mechanism which is what I've tried to do here, it will be a completely overcomeable problem. It's v common, but it's also commonly misunderstood and therefore not optimally treated.
Best of luck