Congress has no authority, however, over the Supreme Court.
Article III , Section 1 of that Constitution reads in part "The judges, both of the supreme and inferior courts, shall hold their offices during good behaviour..."
Since SC Justices, can under this section be removed from their seat if they don't maintain good behavior someone has the authority to deterime if good behavior is being maintained or not.
Article I, Section 2 grants that authority to the Congress "The House of Representatives shall choose their speaker and other officers; and shall have the sole power of impeachment."
(As was exercised against Justice Samual Chase.)
An Inspector General to investigate claims and make recomendations to the Congress on whether or not they should use their authority would seem to fall within the provisions laid out in the Constitution. (I don't agree that one is necessary but I'd disagree that Congress doens't have the authority to create such an office.)