Couldn’t the chairmen then refer it to him, to direct the judge to comply or to be fired?
Well, the Congress could -abolish- that court in an hour and impeach the judge. Since they have THAT power, it stands to reason they should have access to review her work without a separation of powers issue.
I suggest you read Judge Collyer’s responses to Nunes and Goodlatte. It will make things much clearer on the judge’s position. She is not the villain in this case.