The academy has been caught up in the matter since November, when the newspaper Dagens Nyheter reported that at least 18 women had accused Jean-Claude Arnault, a major cultural figure, of sexual assault and harassment.
Mr. Arnault is married to the poet Katarina Frostenson, a member of the academy, and together they run a private cultural club, called the Forum, that has received money from the academy.
The newspaper reported that Mr. Arnault had been accused of mistreating women at the club and at academy-owned properties in Stockholm and Paris over 20 years. It also reported that Mr. Arnault had leaked information about the winner of the prize seven times since 1996.
The Swedish police are looking into the abuse allegations; Mr. Arnault has denied any wrongdoing. His wife agreed on Thursday to withdraw from the academy’s activities.
In the wake of the allegations, Ms. Danius, 56, cut the academy’s ties with Mr. Arnault, and hired a law firm to conduct an investigation into its ties with the private club. The law firm found what it called financial irregularities in that relationship, and recommended that the academy file a police report, but the academy did not take such action.
The divisions within the august body, which is usually known more for literary prestige than for scandal-driven acrimony, erupted into the open last week when three members quit in protest, including Peter Englund, Ms. Danius’s predecessor as permanent secretary.
On the other side are two former permanent secretaries, Sture Allen and Horace Engdahl, who have made lacerating statements in recent days, calling the reaction to the allegations overblown and denouncing Ms. Danius as a weak leader.
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Asked how the members would restore the academy’s reputation, one member, the writer Anders Olsson, said: “We don’t know. But we have to talk to one another much more.”
Complicating matters, there is no provision for the 18 members of the academy to officially resign. Members are elected to life terms, and if they choose to stop participating in the activities of the academy, their seat is left vacant until death.
Even before the scandal, two seats were occupied by inactive members. With the five recent departures, including that of Ms. Danius, the academy will be down to 11 active members — one short of the 12-member quorum needed to elect new members in the event of a vacancy. King Carl XVI Gustaf signaled this week that he would resolve the problem by using his authority to amend the academy’s rules to allow members to resign and be replaced.
In an interview, Bjorn Wiman, culture editor at Dagens Nyheter, said of the scandal: “This is a complete and utter tragedy for cultural life in Sweden.” He added, “The public’s trust for the academy is perhaps below rock bottom.”
“There has always been the consensus that this has been a competent group and that they have judgment, and that the literary judgment is solid,” Mr. Wiman added. “With this scandal you cannot possibly say that this group of people has any kind of solid judgment.”
Ebba Witt-Brattstrom, a professor of Nordic literature at the University of Helsinki, said in an interview that the ouster of Ms. Danius represented a backlash by men.
“What they did was orchestrate a palace revolution, a coup to get rid of her, because she’s too headstrong,” she said. “A headstrong woman is not what they are used to in the Swedish Academy.”
She cited, as an example, the novelist Kerstin Ekman, who stopped taking part in the academy’s work in 1989 in protest over what she called a failure to adequately speak up for the writer Salman Rushdie, who faced death threats.
“Sara Danius really tried to open up and, in a way, to renew the Academy,” said Professor Witt-Brattstrom, who used to be married to Mr. Engdahl. “She made some very good initiatives. She should have had a few more years.”
Svetlana Alexievich, who was awarded the prize in 2015, told the newspaper Svenska Dagbladet, “Those who have broken the rules should be scrutinized.” She said she had learned from Dostoyevsky “that human beings are unreliable creatures, so laws and rules are needed to ensure that this does not repeat itself.”