Short answers, pulled from the story.
The Liverpool Ministry was formed after the assassination of Spencer Perceval, the sitting Prime Minister. The Prince Regent invited Robert Jenkinson, 2nd Earl of Liverpool, to form a government on the 8th of June 1812.
The Liverpool Ministry lasted from the 8th of June 1812 to the 9th of April 1827, a span of nearly fifteen years. Liverpool served throughout as First Lord of the Treasury and Leader of the House of Lords.
Key Cabinet members included John Scott, 1st Earl of Eldon as Lord Chancellor, Henry Addington, 1st Viscount Sidmouth as Home Secretary, Robert Stewart, Viscount Castlereagh as Foreign Secretary, and Nicholas Vansittart as Chancellor of the Exchequer. Henry Bathurst, 3rd Earl Bathurst, served as Secretary of State for War and the Colonies throughout the ministry.
George Canning became Foreign Secretary in September 1822 following the suicide of Robert Stewart, Lord Londonderry, who had previously held the post as Viscount Castlereagh. Canning also took on Londonderry's role as Leader of the House of Commons.
Arthur Wellesley, 1st Duke of Wellington, joined the Cabinet on the 1st of January 1819 as Master-General of the Ordnance, succeeding Henry Phipps, 1st Earl of Mulgrave.
Robert Peel served in several roles across the ministry. He was Under-Secretary of State for War and the Colonies early in the administration and became Home Secretary on the 17th of January 1822, succeeding Henry Addington, 1st Viscount Sidmouth.