Sleep.... is the BEST remedy... for being sick.
It is restorative and actually is when the body's immune system is doing its work and healing.....
Her coughs... from what you describe.. .is due to "Post-Nasal Drip."
This causes coughing... and is dripping from the Sinuses.
A Sinus cold... is hard to get rid of... and IT can go on and on and on and on....
Since she is 5 years old... she can try a Neti Pot... nasal rinse.
You can get it from Whole Foods or any drug store.
MANY Doctors recommend it... to keep the sinuses healthy... and it also prevents... colds.
My Husband who is prone to sinus trouble, uses it... daily. And has since, gets sick less.
My daughter.... from about 4 years old... can use it with no problem, with our standing next to her. She does it when she is sick.
The cold air/weather... can also exacerbate coughs/any lung ailments. I know... I have Asthma. Cold air... makes it way worse.
But your child, does not have asthma. I am just talking about myself.
If I were you... I would really try to get her Sinus issue under wraps.
It does not cause fever... but does create post-nasal drip and ensuing coughing.
ALSO good for preventing colds.... is... to gargle with salt water... several times a day... if suspecting a cold is coming on for just for regular daily routine. I read this, in a health magazine.
You could also try "Little Noses" which is a children's nasal decongestant.
Colds.... last about 7-10 days. Standard.
all the best,
Susan