The tourists crumbled to a fifth humiliating defeat to suffer only the third 5-0 Ashes series sweep in more than a century of the rivalry at the Sydney Cricket Ground on Sunday.
Cook was at a loss to explain what had gone wrong but was adamant he had no intention of walking away from the captaincy.
"If I knew that answer, I would be telling the guys out there rather than telling you guys," he told reporters.
"I am desperate to try and turn it around," added the 29-year-old, who also played in the 5-0 sweep England suffered in the 2006-07 series.
"I feel as if I am the right man to do it. If I'm not and people higher up want a change because they think that's the best way, I have to take it on the chin.
"I have a lot of experience as a player. I have seen a 5-0 before and I've seen the drive and determination which that caused that team then. I know what that takes to do."
Cook said he hoped coach Andy Flower would also remain in charge of the side, but admitted that Saturday's backing for both men from England and Wales Cricket Board chief David Collier was no guarantee in the long-term.
"I was given the vote of confidence from the board which usually means in football terms you have two weeks and then you're on your bike," he said.
"When you strip everything down, every single player now has to go back and have a look at themselves, have a look at their techniques. Have a look at the way they've bowled and start rebuilding again.
"And that hunger has to come from within to do it."
Cook paid tribute to Australia for what he said was a "thoroughly deserved" victory and apologised to England's vocal "Barmy Army" band of travelling supporters.
"As a side we've let the supporters down because we haven't played very well. The results suggest that," he said.
"The work ethic is there, we've just got to go back to basics. Every player will be feeling that they let themselves down because we haven't performed."
Cook said all-rounder Ben Stokes, who made his debut in the second Test in Adelaide before scoring a century in Perth and taking a six-wicket haul in Sydney, and seamer Stuart Broad were the only England players to emerge with any credit.
"There's anger in me and frustration because for whatever reason we haven't played very well and the buck stops with me," he said.
"Only Broady and Stokesy have delivered close to their potential. I think everyone else will be hurting."
- Sports & Recreation
- Sydney Cricket Ground