the Cox brothers employed a conscientious ghost that did most of their work for them.

that did most of their work for them是定语从句。修饰ghost。that在意义上指代ghost，并在从句作主语。

关系代词在定语从句中作主语，只有一下几种情况在非正式语体可以省略关系代词。

1 从句修饰there be句型中的真正主语时：

  There is a man (who/that) asks for you.

2 关系代词为there be句型中的真正主语：

  I had to use anything (that) there was. 

3 This is the man (who) wants to see you.

4 It is the man (who) wants to see you.

其余情况都不可省略在定语从句中作主语的关系代词。

提问者的问题可能是没有看出定语从句导致的。

句子分析：

In time, it became an accepted fact that the Cox brothers employed a conscientious ghost that did most of their work for them.

in time: 时间状语

it：形式主语

became：系动词

an accepted fact: 表语

 that the Cox brothers employed a conscientious ghost that did most of their work for them：真正主语从句

主语从句进一步分析：

that：引导主语从句的从属连词

the Cox brothers: 主语

employed: 谓语

a conscientious ghost: 宾语

 that did most of their work for them：定语从句修饰ghost. 其中that为主语，did为谓语，most of their work为宾语，for them为状语。