I'm really pleased to hear that, OP.
Dig out that asthma management plan, make sure you've got a peak flow meter,( see he's only 5, but this can be an ideal time to get au fait with a PF meter, make it playful, ie, who can get the highest?....don't burst his bubble by blowing to the top, yourself!...)
Getting him used to and familiar with inhalers, spacers...( have you got one?) and peak flow meters will normalise what can become frightening and disrupting future scenarios.
He still young, so HCPs may be a little reluctant to formally diagnose asthma, but it is critical that you watch and be vigilant about how his symptoms behave as he starts school, especially, working with school to find out if he struggles ( ie, coughs, gets short of breath, wheezes, or simply flags) during PE, playtime, outdoor exercise).
Look hard at him at home too, during these activities because exercise and suddenly going out in cold air can trigger these symptoms.
Note how he is after a viral infection, especially cold. Very common to get residual inflammation that causes cough, wheeze, shortness of breath or fatigue after a cold. Nighttime cough is very persuasive of out of control asthma with attendant symptoms and history.
Do note that cough is often the only symptom of asthma; not necessarily wheeze.
With a peak flow meter, it would be great to establish his 'best', ie, measure it morning and night for 2 weeks, when he is well, so you can compare his 'best' measurement with when he demonstrates symptoms.
Do book him in to see an asthma nurse, so it is recorded on his notes that he has needed a 5 day course of oral steroids and to come up with a plan to review him.
He may need to start on a regular 'brown' inhaler in the future, to address ongoing persistent inflammatory airway constriction. But this will need assessment and recording of symptoms, possible peak flow measurement ( if he is deemed old enough) and the taking of a proper history and family history.
Well done, OP.
All the best....it'll take about 12-24 hours of oral steroids and you'll see a real relief of his symptoms. You'll both sleep better.