Nine years ago, Sheila P. Burke was a young nurse with a facility for politics who worked as program director of the National Student Nurses Association in New York. On weekends she filled in as a medical-surgical nurse at Manhattan's Doctors Hospital.

Today, at age 35, Burke is chief of staff for Senate Majority Leader Robert J. Dole (R-Kan.). Her appointment to that position two weeks ago made her, arguably, the most powerful woman in the vast, invisible machine that supports the work of the Senate's 100 visible components. And she is believed to be the first woman to hold the job -- one of the most influential, exacting and exhausting staff positions in Congress.

Burke's Washington career began when she heard from staff members at the Senate Finance Committee, where she had had some contacts through her job with the Student Nurses Association, that Dole was looking for someone with medical experience to be his aide for health issues. She applied for the job and Dole, who has an excellent track record for spotting able women, hired her.

Since then, Burke has moved steadily up the ladder. Past experience paid off in her work on the revolutionary hospital prospective payment system and other momentous Medicare and health program changes as Dole's legislative assistant. Later she became a Senate Finance Committee staff member and deputy staff director, and for the last year she has been one of Dole's top aides in the majority leader's office. Along the way she picked up a Master of Public Administration degree from Harvard's Kennedy School of Government.

Burke won promotion, Senate observers say, by knowing how to track legislation, by knowing how to explain it clearly and rapidly, in a matter-of-fact voice, without histrionics, by having excellent judgment on what will "go" and what will not, and by being remarkably aboveboard and open.

Burke speaks in unhurried, low tones, and strikes a visitor as possessing a calm unusual in the congressional pressure cooker. Those who work with her describe her in recurring terms: an assured command of her subject matter, and the message that anything worth doing is going to get done well. If other Senate staffers utter any criticism of her, it is that a "take-charge" persona includes a flip side -- the message that anything worth doing is best done her way.

But her way of doing things generally draws very high marks. "She's absolutely straightforward, extraordinarily knowledgeable -- in her special areas totally knowledgeable -- and there's no bull about her," said one lobbyist who deals with her often, Jerold Roschwalb of the National Association of State Universities and Land Grant Colleges.

Although she works some on scheduling floor action, her main function, she said in an interview, is following legislation so that Dole knows exactly what is being proposed. "My job is making sure he's prepared," she said. "At the Finance Committee, I always tried to give them the best information I could, not just present one side of an issue."

That may sound simple, but it is more than a matter of memorizing what is in a bill. It is well understood on Capitol Hill that while personal prestige, popularity, a good television appearance and good political action committees are terribly important in getting things done, probably the most important form of legislative clout is knowledge.

Knowledge of the details of an issue. Knowledge of where each group of players stands -- the members of the Senate, Democratic and Republican, the president, the House, the interest groups -- "what they see as their priorities." Knowledge of which of their positions are immutable and which are open to compromise. The kind of knowledge that enables a Senate leader to make instant decisions and craft legislative compromises.

To provide that, Burke said, it is necessary to have a staff culling out details, consulting experts and talking to spokesmen for different groups and different players in the legislative game.

Being legislative chief of staff means working a very tough, long day, usually beginning at 7:30 a.m. A typical day can start with a breakfast -- maybe with lobbyists, maybe with Senate staff, maybe with agency officials -- and end 13 hours later. "The most difficult part of the job," Burke observes, "is keeping track of the number of things going on at one time."

Every Friday morning, she and her colleagues in the speaker's office meet with the staff directors of the Republican-controlled Senate committees, just "to see what is on their minds and those of the members and committee chairmen they work for."And she talks "all the time" to White House legislative staff, and to David J. Pratt and Scott Bunton, top aides to Minority Leader Robert C. Byrd (D-W.Va.).

The start of Burke's new job coincides with what is shaping up as one of the busiest -- and, some say, the most contentious -- sessions in memory, with control of the Senate up for grabs this fall, and with major tax and budget issues on the docket. Burke has started mastering the details of the president's budget and trying to find out from the various GOP committee chairmen what they propose, and how they hope to deal with spending and deficit problems on matters within their jurisdiction. From that, she has assembled a report to Dole on what the various committee chairmen would like to see in the fiscal 1987 budget.

She also has started working on the budget reconciliation bill that failed to pass late last session because it included a manufacturing excise tax. She is trying to find out if some of the bill can be salvaged.

Burke was born in San Francisco and moved to the town of Merced, in the San Joacquin Valley, when she was 10. She went to college at the University of San Francisco, a Jesuit institution, and studied nursing. Then she worked as medical-surgical nurse in Alta Bates Hospital in Berkeley for a year before moving to New York.

Although Burke has had many dealings with White House staff in the past and expects to have plenty more, one staffer she does not have much professional contact with is David Chew, a high-level official and deputy assistant to the president who is her husband. They were married three years ago.

Chew, a one-time aide to Dole and then to White House chief of staff Donald T. Regan when Regan was secretary of the treasury, "controls the flow of paper to and from the president," according to White House officials -- reports, proposals, lists of policy options, analyses, orders.

Undoubtedly, those papers would tell Dole a lot if he saw them. And Burke could surely pass along plenty of information about Dole's plans and strategies that the White House would like to have. But Burke says of her husband, "I rarely deal with him at the White House. I deal with the legislative staff for Capitol Hill."

This potential conflict is an interesting problem that Burke has in common with her boss, who is married to Transportation Secretary Elizabeth Hanford Dole. Burke says plainly that she and her husband do not exchange information or constitute a back channel between Dole and the White House. "We keep our secrets," she says with a smile.