Who vs. that

I have a question that's been rather difficult to research using Google, so hopefully someone here would be kind enough to address it.

I am an American English teacher in Japan, and one of my colleges (a Japanese English teacher) corrected my usage earlier.

The grammar in question is as follows: "Who is the boy who is sleeping?"

I maintain that the grammar is fine, perhaps even preferable, as using "that" to refer to a person is kind of a shaky matter; my colleague says that one should change the second "who" to "that," as the word "who" was already used at the beginning of the sentence.

What do you all think?