In what language did Hezekiah's officials ask the Rabshakeh to speak to them?

Answer: Aramaic

Eliakim, Shebna and Joah asked him to speak Aramaic (KJV: "the Syrian language"), not Hebrew in the hearing of the people on the wall.

“Then Eliakim, Shebna, and Joah said to the Rabshakeh, “Please speak to your servants in Aramaic, since we understand it. Do not speak to us in Hebrew in the hearing of the people on the wall.””— Isaiah 36:11 (BSB)