Harry Potter's creator brought proceedings in London's High Court against Chris Gossage, a partner at Russells and a friend of his, Judith Callegari.
Her solicitor, Jenny Afia, told Mr Justice Tugendhat that Rowling was revealed in the Sunday Times as the writer of crime novel The Cuckoo's Calling, which was published under the name of Robert Galbraith.
A few days later, Russells contacted her agent disclosing that it was Mr Gossage who had divulged the confidential information to Ms Callegari, who then communicated it in the course of a Twitter exchange with a journalist.
Ms Afia said that Ms Rowling, who was not in court, "has been left dismayed and distressed by such a fundamental betrayal of trust".
Mr Gossage, Ms Callegari and Russells all apologised, with the firm agreeing to reimburse Ms Rowling's legal costs and make a payment, by way of damages, to the the Soldiers' Charity, formerly the Army Benevolent Fund.
She said: "This donation is being made to The Soldiers' Charity partly as a thank you to the Army people who helped me with research, but also because writing a hero who is a veteran has given me an even greater appreciation and understanding of exactly how much this charity does for ex-servicemen and their families, and how much that support is needed.
"I always intended to give The Soldiers' Charity a donation out of Robert's royalties but I had not anticipated him making the bestseller list a mere three months after publication - indeed, I had not counted on him ever being there! "
Major General Martin Rutledge, chief executive of the charity, said: "We are absolutely thrilled by the extraordinary generosity of JK Rowling who is such an internationally renowned author.
"This donation will make a huge difference to the lives of thousands of soldiers, former soldiers and their families who are in real need.
"Her tremendous show of support for The Soldiers' Charity will help to remind people of the many sacrifices made by our soldiers, long after any news of Afghanistan has left the front page."
Ms Afia told the court that the author used the pseudonym to release the book in April free from the unprecedented publicity and expectation that accompanied her work.
Only a handful of trusted advisers, family and friends were aware of Galbraith's true identity until the front-page story made headlines worldwide.
"The claimant was angry and distressed that her confidences had been betrayed and this was very much aggravated by repeated speculation that the leak had, in fact, been a carefully co-ordinated publicity stunt by her, her agent and her publishers designed to increase sales.
"The claimant has been left dismayed and distressed by such a fundamental betrayal of trust.
"As a reflection of their regret for breach of the claimant's confidence, including frustrating the claimant's ability to continue to write anonymously under the name Robert Galbraith, the defendants are here today to apologise publicly to the claimant."