Final answer:
If the U.S. House of Representatives fails to elect a President, the Vice President-elect acts as President. If there's no Vice President-elect, Congress has the authority to decide who acts as President, typically falling to the Speaker of the House then onward through the Presidential line of succession.
Step-by-step explanation:
If the election of the President of the United States becomes the duty of the House of Representatives and it fails to act by the time fixed for the beginning of the President's term, then the Vice President elect shall act as President until a President qualifies. However, if there is no Vice President-elect, Congress has the authority to provide by law for who shall then act as President, or how one who is to act shall be selected.
This situation is guided by the 20th and 25th Amendments, as well as the Presidential Succession Act. In a case where neither a President-elect nor a Vice President-elect is qualified, the Speaker of the House is next in line to act as President, followed by the President pro tempore of the Senate, and then by Cabinet members, in the order of their department's creation.
In summary, the Constitution and subsequent laws ensure a continuity of government by defining a clear line of succession beyond just the Vice President, including the Speaker of the House and other officials within the U.S. government.