the irony being that she was widely understood to have spoken out aginst it in the privacy of the committee.

这不是一个句子，而是一个“独立主格结构”。其构成为：

逻辑主语（the iroy）  + being that....（现在分词短语）

其中，现在分词短语是“系表结构”，且表语由从句充当。

如果改为句子，则是：

The irony is that she was widely understood to have spoken out aginst it in the privacy of the committee.

【翻译】具有讽刺意味的是，人们普遍认为她曾在委员会的私下场合对此表示过反对。