A change in the English language during the 1800s made the phrase the fact that necessary in certain grammatical situations. You should use it in these situations, although it's probably better to avoid it when it's unnecessary.
One can see from Google Ngrams that this phrase was hardly ever used before 1800, and that its use increased steadily throughout the 19th century, peaking around 1920 (roughly when Strunk wrote the book, so possibly his advice had some effect).
Strunk originally wrote the book in 1918, and he was born in 1869. Thus, over his lifetime, he saw the phrase "the fact that" receive more and more use; I assume that he perceived this phrase as ugly, and that this increase in its use bothered him. Today, since it's become so common, I believe that hardly anybody is bothered by this phrase today.
In 1846, one could write
He was an Irishman by affection as well as by the accident of birth, and despite that he was born and bred amongst the aristocracy, had a heart for his country
Today, we would either need to write
and despite the fact that he was born and bred amongst the aristocracy, he had a heart for his country
or rephrase the sentence substantially.
Similarly, in 1878, Anthony Trollope could write
I did not like that you should be in London without my seeing you.
Again, today you would need to insert "the fact that" before "you should" or rewrite the sentence.
Thus, my recommendation is that you leave the sentence as is.
One thing to note: the fact that we now use "the fact that" so often means that people sometimes tend to use it when it's not needed at all. In these cases, you should delete it, as it's redundant. For example, to use an example from Strunk and White,
I was unaware of the fact that ...
should be replaced by
I was unaware that ...
unless you have a good reason for emphasizing that the thing you were unaware of is a fact.