In the last week, Senator Pat Roberts, Republican of Kansas, a longtime friend of the Sebelius family, said she should take responsibility for the technical problems that have thwarted millions of people eager to buy insurance through the federal exchange that opened on Oct. 1. He accused Ms. Sebelius of âgross incompetence,â saying, âWe need new leadership.â  Representative John Fleming, Republican of Louisiana, said Ms. Sebelius should resign or be fired. âTaxpayers should not have to tolerate this kind of waste and incompetence,â he said.  But in an interview, Ms. Sebeliusâs older brother, Donald D. Gilligan, said, âI donât think you resign in the middle of a fight.â  Moreover, Mr. Gilligan said: âThe fact that people are calling for her head does not surprise her or alarm her particularly. People have been calling for her head for a long time, and itâs still there. I donât think that fazes her much.â  Mr. Gilligan said Ms. Sebelius was âjust digging in on a daily basis and trying to figure out how to fixâ the problems that have riddled the insurance exchange.  The secretaryâs sister, Ellen M. Gilligan, said, âTo my knowledge, she is not going to resign,â despite the âwishful thinkingâ of some Republicans.  âShe serves at the pleasure of the president,â she said. âIf the president decides that it is her responsibility and that she is ultimately responsible for the failure of the exchanges, then, of course, she will resign.â  But Ms. Gilligan said that was unlikely.  âThe White House is smart enough to know that if she steps aside or they ask her to resign, they will never get anybody else confirmed,â Ms. Gilligan said. âPlus, I donât think they hold her responsible.â  Discussing the new online insurance market during a visit to Cincinnati on Wednesday, Ms. Sebelius acknowledged the problems with the exchanges, but said that fixes were in the works.  âI am the first to admit that the launch was rockier than we would have liked,â she said. But after two weeks, she added, âthere are vast improvements,â and people who were frustrated should âcome backâ and try again.  Some consumers who did so said they were still unable to log in and shop for insurance on Wednesday.  Mr. Roberts was one of nine Republican senators who voted to confirm Ms. Sebelius in 2009. From 1969 to 1980, he was an aide to her father-in-law, Keith Sebelius, a Republican who represented western Kansas in the United States House of Representatives.  Robert Gibbs, the former White House press secretary, said Monday on MSNBC that the debut of the insurance exchange had been âbungled badlyâ and had been âexcruciatingly embarrassing for the White House and for the Department of Health and Human Services.â After the problems are fixed, he said, âI hope they fire some people.â  But Jay Carney, the current White House press secretary, said Ms. Sebelius had âthe full confidence of the president.â  For months, Ms. Sebelius had been projecting optimism and confidence about the exchange, built by her department, with help from dozens of contractors. âI think weâre on target,â she said in a television interview in July, adding, âWe are on track to flip the switch on Oct. 1 and say to people come on and sign up.â  A new study by a market research company suggests that traffic on the federal Web site has decreased since Oct. 1 and that relatively few people have signed up for health plans.  The number of visitors to the Web site declined to 4.1 million in the second week, from 9.5 million in the first week, said Matthew A. Pace, a vice president of the company, Millward Brown Digital, a unit of WPP, a holding company for advertising and market research agencies.  In the first week, he said, 196,000 people began to enroll, and 36,000 completed the process. In the second week, he estimated, 368,000 began enrollment, and 47,000 finished the process.  The Obama administration has not yet disclosed enrollment data.
Kathleen Sebelius, the health and human services secretary, will not bow to Republican demands to resign after the troubled rollout of the health law, people close to her said.