Update 8.35pm: Karen Bradley has been appointed as Northern Ireland Secretary, moving from the Department for Digital, Culture, Media and Sport, Downing Street said.
Speaking tonight MS Brady said she was keenly aware that there are immediate challenges.
"It is now a year since Northern Ireland has had an effective, functioning power-sharing administration, and forming a Northern Ireland Executive, to deliver for the benefit of all, is my top priority.
"I believe a devolved government in Belfast is best placed to address these issues and take the key decisions which affect people's day to day lives, whether these relate to the economy, public services or issues of policing and justice."
Update 2.39pm: Foreign Affairs Minister Simon Coveney phoned Mr Brokenshire while on a visit to Cairo to wish him well.
Mr Coveney said he had shown unfailing dedication and determination to secure political progress consistent with the objectives and commitments of the Good Friday Agreement.
"His unwavering commitment - in public and in private - over the last year to securing the effective operation of the devolved power-sharing institutions in Belfast has been hugely important," he said.
"While it is not always obvious to the public gaze, very important progress has been made on significant issues over the last year and I believe that a positive outcome can still be achieved. If it is, it will be a testament to the quiet, understated but hugely valuable work of James Brokenshire."
Mr Brokenshire said the parties in Northern Ireland have gotten over bigger issues in the past.
He said: "I think that there is a duty and responsibility on them now to get back into an Executive, back into devolved government and get on with serving the people of Northern Ireland.
"It has been a huge honour and privilege to be able to serve as Secretary of State for Northern Ireland.
"I have been hugely moved and touched by so many people that I have met and seen, worked with in Northern Ireland and that underlines to me the really positive focus that we should have now."
Update 1.48pm: Brokenshire accepted he could not give 'energy' needed in Northern Ireland role
James Brokenshire quit as Northern Ireland Secretary for medical reasons as he acknowledged that forthcoming surgery would mean he could not give the "effort, energy and complete focus" needed for the role.
In his letter to UK Prime Minister Theresa May, Mr Brokenshire said an operation to remove a small lesion in his right lung meant he had to stand down.
Efforts to restore the power-sharing administration in Northern Ireland and the impact of Brexit on the island of Ireland mean that the role is demanding and sensitive and Mr Brokenshire said he had hoped to lead the "essential work with renewed intent" before his diagnosis.
In his letter to Mrs May he said: "I recognise that this comes at an important moment for politics in Northern Ireland."
There was an "urgent need" to restart stalled talks on the restoration of devolved government, he said.
"We are now well into overtime to re-establish an executive if further intervention in the day to day affairs of Northern Ireland is to be avoided."
Mr Brokenshire, 50, said he had been informed about the lesion "in the last few days" after a series of tests in recent weeks.
He said his decision to step down was the right thing to do.
"I have a small lesion in my right lung that will require surgery to remove. Whilst the health team believe that will deal with the issue and that I will be back to work - it will take a number of weeks," he said.
The Prime Minister appeared to hold out the prospect of a return to government for Mr Brokenshire, who had previously served under her in the Home Office with responsibility for security and immigration.
Mrs May said Mr Brokenshire had demonstrated that the role in the Northern Ireland Office was "vital work which will demand long hours, hard effort and complete focus" and it was "absolutely right that you should put your health first".
She told him he had performed with "great diligence, determination and good humour" in his government roles and "I know that you will approach your forthcoming operation in the same way".
She added: "I very much look forward to working alongside you again when you are back to full health."
Mrs May sent her best wishes to Old Bexley and Sidcup MP Mr Brokenshire, his wife Cathy and their three children.
"While it is typical of you that your first thought was not for yourself, but for your duties as a Cabinet Minister and public servant, it is absolutely right that you should put your health first, for your sake and that of your family," she said.
DUP leader Arlene Foster said: "Since becoming Secretary of State in 2016, Mr Brokenshire had immersed himself fully in the role by dedicating long hours to trying to make progress.
"James leaves the role with a very intimate knowledge of Northern Ireland and I look forward to working with him again in the future."
Earlier: James Brokenshire resigns as Northern Ireland Secretary due to ill-health
Northern Ireland Secretary James Brokenshire is understood to have resigned from the UK Cabinet position due to ill-health.
Northern Ireland Secretary James Brokenshire has quit the Cabinet in the first of what are expected to be a number of changes as Theresa May reshuffles her top team.
The Northern Ireland Office confirmed his resignation, but gave no further details.
However, a source close to the 50-year-old minister said he had decided to stand down because he was facing major surgery within the next couple of weeks.
The Old Bexley and Sidcup MP is a close ally of Mrs May, having served under her for five years at the Home Office, and he was not among ministers who were predicted to go in the UK Prime Minister's first major reshuffle since she took office.
Explaining his decision, a source close to Mr Brokenshire said: "He has a small lesion on his right lung and is getting major surgery in the next couple of weeks."
The Prime Minister will seek to stamp her authority with a Cabinet reshuffle beginning today amid reports that up to six senior ministers could be axed or moved.
As MPs prepared to head back for Westminster following the Christmas break, she reaffirmed her intention to lead the Conservatives into the next general election.
Downing Street sources indicated the reshuffle was expected to be conducted over two days, with junior and middle-ranking ministerial appointments likely to continue into Tuesday.
It is likely to represent her biggest overhaul of her top team since she appointed her first Cabinet on entering No 10 in 2016.
She made only limited changes among her senior ministers following the election in June having seen her position badly weakened by the loss of her overall majority in the Commons.
It is thought her most senior ministers - including Chancellor Philip Hammond, Foreign Secretary Boris Johnson, Home Secretary Amber Rudd and Brexit Secretary David Davis - will remain in their current posts.
However, Education Secretary Justine Greening, Conservative Party chairman Sir Patrick McLoughlin, Business Secretary Greg Clark and the Leader of the Commons Andrea Leadsom are among those reported to be vulnerable.
Downing Street sources sought to play down the reports, describing them as "speculation" and "guesswork".
It is thought that Mrs May will take the opportunity to bring forward some more junior ministers, with Immigration Minister Brandon Lewis and Justice Minister Dominic Raab among those tipped for promotion.
It is unclear, however, whether she will announce a direct replacement for Damian Green who was forced to quit as her effective deputy after he admitted lying over the alleged discovery of pornographic material on his Commons computer during a police raid in 2008.
Health Secretary Jeremy Hunt had been widely seen as the favourite for the post, although reports have suggested that she may be reluctant to move him in the midst of an NHS winter crisis.
Shadow health secretary Jon Ashworth said yesterday: "If she promotes this Health Secretary tomorrow it's a betrayal of those 75,000 people in the back of ambulances."