***** NOT A TEACHER *****
I am 95% confident that you are right.
Let's add the optional "that":
"All I know is (that) he fell down the stairs."
This "that" is not a relative pronoun. It just introduces the noun clause: "The fact is (that) you are an excellent student."
Now let's reverse the order:
"That he fell down the stairs is all I know." (This time, you MUST use the "that.")
Another test that teachers often give for noun clauses is this: If you substitute the word "something" for the noun clause, will the sentence still be grammatically correct?
Mona: All I know is something.
James: Really? What is that "something"?
Mona: That he fell down the stairs.
Thus, I believe that "All I know" is the complete subject. / "is" is a linking verb./ "(that) he fell down the stairs" is a subjective complement because it "completes" the meaning of the subject. Of course, you could NOT say to someone: "All I know is."