I spoke briefly about Martin Van Buren's time as Vice-President here. To my knowledge he is the only Vice President to wield a significant amount of political power within an administration from the founding to the Civil War. Jefferson would also meet your criteria, but it should be noted he was a Vice-President opposing the political administration of Adams. Jefferson's political power thus came from his role as leader of the Republican party, and he wielded little influence within the administration.

I should clarify in that Vice Presidents often continued to wield considerable political influence in their own native state but had little to no influence within the Administration. I feel this distinction is important as many Vice Presidents would meet your criteria but had no political power in the administration in which they served.

Richard Nixon is an odd figure here. He was, for all intents and purposes, absolutely integral to Eisenhower's ability to lead. Nixon was popular in the West, wildly popular with conservative Republicans, and had a strong anti-communist while not crazy offensive (McCarthy) record. At the same time, Eisenhower quite clearly disliked and unarguably did not respect Nixon...for the very fact that Nixon held and sought considerable political power.

What's interesting is that the Eisenhower-Nixon(-Dulles) dynamic saw the two elders constantly teaching and challenging Nixon, though often by effectively smacking him down every time he attempted to act on a previous lesson (thus teaching him a new lesson).

Still, as VP, Nixon had some strong influence on Civil Rights (though Eisenhower's active disinterest in the issue stunted progress), foreign policy in Latin America (however inadvertently), and in Africa (despite Dulles's best efforts).