Please do not let me dampen your hopes, but I did not see much sign with my brother that passive exposure to language (videos and TV) made much of a difference in his speech skills. It did improve his language comprehension, though.
Sign was a big help in our family, if only because that broke down a communications barrier. Call it a crutch if you want, but perhaps such a crutch is nice when it makes it possible to talk! 
A hearing aid helped even more. (It was discovered late that Derek is mildly hearing impaired.)
Derek has grown out of sign with time. Now if I don’t understand him and I ask him to sign, he tells me, “No. Too slow.” I usually sigh and let him teach me what he is saying.
