It's kind of confusing since the LOC serves Congress, but the Librarian of Congress is a President-appointed and Senate-confirmed position. They're supposed to serve for 10-year terms (she was appointed in September 2016) though, and my understanding is it's a open question whether the President can legally fire a LOC before their term is up.
I think the "senate-confirmed" part is important and relates to three president's inability to just install a replacement independently. He can remove an official nominated by the president, but he can't unilaterally fill the vacancy (even temporarily ) for positions outside of the executive.
https://www.law.cornell.edu/uscode/text/2/136-1?hl=en-US