WASHINGTON (Reuters) - When U.S. President Barack Obama accepted the resignation on Friday of Veterans Affairs Secretary Eric Shinseki, he said his priority now was fixing the troubled agency whose officials are accused of covering up delays in providing healthcare for U.S. veterans.

That task is monumental. As Obama himself said, the sprawling Veterans Affairs department "has had problems for a very long time," including management problems. The agency's woes have been compounded by the rising demands for services after the return of veterans from Iraq and Afghanistan. Obama noted on Friday that the VA enrolled 2 million new veterans in healthcare under Shinseki's watch.

Obama and many Democratic lawmakers say that the increase calls for more doctors and nurses to prevent veterans from having to endure long wait times for care.

But hiring them would require a big funding increase from the U.S. Congress, something that is unlikely to happen given the strong resistance in the Republican-led House of Representatives to spending increases.

In February, Senate Republicans blocked a bill by Bernard Sanders, an Independent who caucuses with Democrats and chairs the Senate Veterans Affairs Committee, to expand veterans' benefits. Just two Republican senators voted for it. It would have cost about $21 billion over a decade.

Representative John Culberson, Republican chairman of the House appropriations subcommittee in charge of veterans affairs, said the VA is already "amply funded. The problem has far more to do with bureaucratic rigmarole and civil service inertia than it does with lack of funding." He noted that the VA has been exempt from automatic spending cuts that were imposed on other agencies starting last year as a result of a budget deal between the two parties.

Proposals for broad legislative changes to the VA could also fall victim to gridlock, especially in a midterm election year when both parties are more focused on attacking one another than on striking bipartisan compromises.

On Friday the Obama administration moved to address one management practice that may have given some VA administrators an incentive to falsify records - a bonus system that rewarded officials for shorter wait times. The administration is doing away with all performance bonuses this year for senior VA health executives.

To ensure that veterans aren't left waiting for doctor appointments and urgent care, both Republicans and Democrats have suggested offering more patients access to private healthcare services, at least on a temporary basis. Sanders, however, says that would not work in the long term because there are not enough doctors in the private sector either.

Obama tapped Deputy VA Secretary Sloan Gibson to serve as acting head of the Veterans Affairs department and has promised to look for a permanent successor to Shinseki.

Gibson, an Army Ranger and former banker, joined the agency three months ago and is a former head of United Service Organizations Inc, a nonprofit that assists veterans and their families.

Some veterans advocates and lawmakers believe he would make a strong leader of the VA.

Culberson praised Gibson as "deeply passionate about making sure these veterans are taken care of" and said his background in both the military and corporate world would be a plus.

Other possible candidates that have been mentioned include former Army Chief of Staff Ray Odierno, former Virginia Senator Jim Webb and Representative Tammy Duckworth, a Democrat who lost both legs in an Iraq war helicopter crash.

Senator Jack Reed, a Democrat and leading member of the Armed Services committee, said one way to get around partisan differences over fixing the VA would be to have a "top to bottom" review of the VA system by outside experts.

"Ultimately, if you have a thorough review that is seen as legitimate and independent, those recommendations are likely to be followed by both sides of the aisle," Reed said in an interview.

Sanders views a lack of resources as one of the VA's biggest problems. He plans to introduce legislation that would encourage more primary care doctors and nurses to work for the VA.

His bill would offer scholarships and student loan forgiveness programs that would encourage students to take jobs at VA medical centers and clinics.

Representative Adam Smith, the senior Democrat on the Armed Services committee, said the healthcare agency needed more money. After over a decade of war, "the VA has more and more veterans needing help."