The President, in this case, Barack Obama, appoints the Librarian of Congress. Here's the statute about that particular appointment.
Should it be a presidential appointment? Should the next office holder have a degree in Library Science (Mr. Billington did not).
Infodocket has more information on the period of transition, including this:
LC tells us that while no timeline is in place at the moment, President Obama has “roughly” six months to consider nominees for the vacancy. If a new Librarian of Congress is not confirmed by the time of Dr. Billington’s retirement, David Mao, Deputy Librarian, would serve as Acting Librarian of Congress until the time a new leader is confirmed by the Senate. Mao holds both legal and library degrees.