Thanks for your query.
The cough your mother has is due to eosinophilia
as the count is 14 so hetrazan
should help her get better.
Since she is 58 yrs she is probably having osteoarthritis
of knee which is common at her age.
Some pain killers are not to be used in people who have history of asthma
so you have to see if she had history of asthma.
The other reason why a pain killer can cause increased cough is due to increase in acidity there can be reflux resulting in increase of cough.
As she has fever
it will be advisable to consult her doctor and get her examined and if he feels a need to start on antibiotics he will do so.
Hope this answer is helpful.