I finally started to recover from my cough which was an aftereffect of my infection the last few days.
この場合、次のような言い方ができますよ。
ーI finally started to recover from my cough which was an aftereffect of my infection the last few days.
「ここ数日やっと感染症の後遺症の咳から回復してきた」
I finally started to recover from ... the last few days. で「ここ数日やっと…から回復してきた」と言えます。
ご参考まで!