To some analysts, Kane's letter raised doubts that she will ever hand over the case.
And should she turn it over, those analysts say, her dare could boomerang.
Among the possible scenarios: a successful prosecution by Williams, or a media circus in which the district attorney makes public dramatic audio and video of legislators taking money.
"If the file contains evidence that the sting operation was working when it was shut down, that would be very embarrassing to her," said L. George Parry, a Philadelphia defense lawyer and veteran former prosecutor of official-misconduct cases.
G. Terry Madonna, the political analyst and pollster, said Kane miscalculated in making the challenge.
"What happens if he decides to prosecute and gets a conviction? It would be a huge political setback for her," Madonna said. "It seems to me she has got to find a way out of this."
The sting began in 2010, when Republican Tom Corbett, now the governor, was the attorney general, and continued until early 2012. It caught five Philadelphia Democrats on tape taking money or gifts, according to interviews and court records.
Kane, a Democrat, said her review of the case found it "not prosecutable," poorly run, and possibly tainted by racial profiling. The five elected officials caught on tape are African American.
In an unusual public feud between top Democrats and top prosecutors, Kane made her challenge to Williams hours after he had sharply criticized her in an Inquirer story published April 9. Williams accepted the challenge almost immediately.
After two weeks of silence, Kane wrote back Wednesday. In a letter to Williams, a Kane aide gave him a taste of what Kane has been enduring in recent weeks - raising possible conflicts of interest.
Frank G. Fina, a former top state prosecutor who now works for Williams, had contended that Kane should play no role in the sting case because the informant in the investigation had had dealings with two political supporters of Kane's. Kane has said she faced no conflict.
In the letter to Williams last week, Kane's office said the district attorney could face a conflict in taking on the case because he was endorsed by two of the five Philadelphia Democrats who allegedly accepted money.
The letter also raised another potential conflict: that Williams knows the sting's undercover operative, Tyron B. Ali.
Investigative documents show that Ali and Williams attended a political party on New Year's Day 2011 at the West Philadelphia home of State Rep. Louise Bishop. Bishop is one of the officials who sources say later accepted cash from Ali.
At the party, Williams and Ali spoke briefly, the documents say. Ali later said Williams had told him he had not seen him in a cigar shop lately, and that perhaps they would run into each other there soon.
On Friday, Williams wrote to Kane and accused her of making "unsound" claims and empty demands.
He criticized Kane for citing his brief meeting with Ali.
"The implication you create is that a casual exchange of pleasantries may have amounted to some sort of illegality," Williams wrote in the letter, obtained by The Inquirer. "I believe you know that any such implication is false."
Williams said he would continue to press for the case file in the sting investigation.
Larry Ceisler, a Democratic communications consultant in Philadelphia, said Kane had built up a very positive image until the sting controversy. He said a reversal on turning over the case file could exact more damage.
"If she pulls back, it undercuts a reputation for decisiveness," he said. "That's what you want in a prosecutor."
Some legal experts said Kane may have overreached in suggesting that elected law enforcement officials face conflicts when pursuing cases against people who supported their campaigns.
Anne Poulin, a law professor at Villanova University and an expert on how prosecutors deal with conflicts of interest, said they invariably face vexing choices when they conduct investigations that involve political allies.
But she said Williams had defused any ethical questions by pledging to review whether the five Democrats captured on tape could be prosecuted.
"If that kind of case comes across his desk and he says, 'I don't care - I am prosecuting them,' then you don't have the appearance of impropriety," Poulin said.
Bruce A. Green, a law professor at Fordham University and a former federal prosecutor, also he doubted that the district attorney faced a conflict.
"It can't be that the prosecutor has to recuse himself any time he investigates somebody who has supported him or opposed him," Green said.
Kane has been adamant that she, and not Fina, should have been the one to decide if she faced a conflict in the sting case.
Green agreed with Kane on that. "He wasn't elected," he said. "She was elected."