Here are the rest of Vien's questions. They are all on the same grammar point, and can be addressed comfortably in one posting, so I am posting these questions here. Vien wants to know if the new sentences are correct.

sentence 3/ emphatic "A noun clause can be a subject complement"
Billy's mistake was that he refused to take lessons.
-> It is that he refused to take lessons that/ which Billy's mistake was


sentence 4/ emphatic "A noun clause can be an object of a preposition"
Mary is not responsible for what Billy did.
-> It is what Billy did which/ that Mary is not responsible for

sentence 5/ emphatic "A noun clause (but not a noun) can be an adjective complement"
Everybody is sad that Billy drowned.
->It is that Billy drowned which/ that everybody is sad.

sentence 5/ emphatic "Noun phrase as subjects of verbs"
-> how to swim is a miracle.
It is how to swim which/ that is a miracle.

sentence 6/ emphatic "Noun clauses as objects of verbs"
I don't know where to go.
-> It is where to go which/ that I don't know.

sentence 7/ emphatic "Noun clauses as subject complements"
The question is how to try the same thing.
-> It is how to try the same thing whch/ that the question is.

sentence 8/ emphatic "Noun clauses as adjective complements"
He is happy to learn English.
-> It is to learn English which/ that he is happy.

from Vien Phan

In the sentences you are asking about, Vien, here are my comments:

"¢ 3) It is the fact that he refused to take lessons that / which was Billy's mistake.

(Note again that "it is the fact that" is used. And, note the word order of the last clause: "that / which was Billy's mistake.")

4) The sentence is OK.

"¢ 5) It is the fact that Billy drowned that / which has made everybody sad.
or
"¢ It is the fact that Billy drowned that / which everybody is sad about.

"¢ It is the fact that people know to swim which/ that is a miracle.

6) OK

7) OK

"¢ 8) It is learning English which/ that he is happy about.
or
"¢ It is the fact that he is going to learn English that he is happy about..

Rachel