There is a simple solution here short of the disruptive impeachment process. Simply defund the VP's office. Leave enough money for his salary and a secretary, and let him live in the VP's home (maintain that for the next VP). However, do not let him have any staff, travel at government expense, or do much of anything else beyond the constitutional duties of the VP.

Defunding would be humiliating (good!) and simple, as the House could easily do it. There should be enough votes in the Senate to approve this in the next budget. Defunding would also prevent most, but not all, of Cheney's tricks.