If we want to stop Cheney, we should stop funding him. This is simple. The next procurement bill can have a short paragraph stating that "The Office of the Vice-President shall receive no funds beyond those required by law to pay the Vice-President's salary."
If we want to make it rock-solid, let's add: "No funds shall be diverted from other purposes to fund any functions of the Vice-President's office, whatsoever."
This will close Cheney's "office," and some of his power. Don't even give him a secretary. Force him to get his own coffee, make his own copies, and represent himself to the press. The Constitution does not stipulate anything beyond that his salary can't be lowered during his term of office.
This can be done swiftly, and the Congressional Republicans will be hard-pressed to defend the staffing of an office that has lied to them and to the American public.
Attach this to each and every spending bill. Let them filibuster or veto it: Cheney becomes the isssue either way. There is NO downside for Democrats.