One big confusion here; 使わせる(tukawaseru) is causative, not passive. Your translation "that kid that is making him speak using keigo" is correct.
Your new sentence "anna otoko ni keigo wo tsukawaReteru ano ko..." is correct, too. The only difference from the original sentence is whether the boy is making the chief guy speak in keigo, or the chief is speaking in keigo probably voluntarily.
"The kid that is being forced by that man to speak using keigo" should be "anna otoko ni keigo wo tsukawaSERAREteru." This is passive voice of causative sentence.
Btw, "the chief" is "kashira," not "atama."
They are same in kanji, but just different readings.