He made eyes at her, was taken with sudden coughs and 'hems' ...
1. be taken with：为/被......所吸引。系表结构。又如：
She seems very taken with the idea. 
她好像对这个想法很感兴趣。
《柯林斯英汉双解大词典》
I think he's quite taken with the idea. 
我认为他对这个想法十分感兴趣。
《牛津词典》
We were all very taken with his girlfriend. 
我们都觉得他的女朋友非常讨人喜欢。
《牛津词典》

2. be taken with：被......所侵袭；引申为 “突然发作”、“突然爆发出”。本句的意思便是。