Creating an executive agency to investigate the executive would be pretty stupid. It'll likely lead to "we've investigated ourselves and found nothing", even if the president wasn't vested in all executive power, as the executive branch consolidates.

However, creating an agency in the legislative branch is another story. One of the duties of the legistive is to investigate the executive, is it not? (not that this matters when both the executive and legislative are controlled by the same party)

The GAO is a "legislative agency"...