Sentence Correction

A and C can be eliminated for using present perfect (has waited) for an event that happened in the past.

D can be eliminated for using "was waiting".

As you've identified, the only difference between B and E is the word "that." In this case, we need it because "his client was not going to show up" is a dependent clause. E is the better option.

This is a good example of a fairly common situation on Sentence Correction: two answers that are almost identical. When that happens to me, I like to "reset" and focus on the difference(s) between the two.

Bill