Answer: The Senate.
Explanation: To balance the role of the House in breaking presidential ties, the Twelfth Amendment requires the Senate to handle that Job for tied vice-presidential contests. The Senate must choose between the two top electoral vote recipients, with at least two-thirds of the Senate's members voting.
The Great Compromise was a compromise between the New Jersey Plan and the Virginia Plan.
The New Jersey Plan, basically, was for each state to get an equal vote in Congress, while the Virginia Plan, again, briefly, was for each state to get a vote proportional to the amount of citizens living in their state.
The Great Compromise solved the concerns that the large and small states had, of not getting enough power in Congress. They compromised, and created the House of Representatives, following the Virginia Plan, and created the Senate, which is above the House of Representatives, following the New Jersey Plan.
