What is the full name of the person who helped Charles win the design contest for the new palace?
The Palace of Westminster, the medieval royal palace used as the home of the British parliament, was largely destroyed by fire on 16 October 1834. The blaze was caused by the burning of small wooden tally sticks which had been used as part of the accounting procedures of the Exchequer until 1826. The sticks were disposed of carelessly in the two furnaces under the House of Lords, which caused a chimney fire in the two flues that ran under the floor of the Lords' chamber and up through the walls.
The resulting fire spread rapidly throughout the complex and developed into the biggest conflagration in London between the Great Fire of 1666 and the Blitz of the Second World War; the event attracted large crowds which included several artists who provided pictorial records of the event. The fire lasted for most of the night and destroyed a large part of the palace, including the converted St Stephen's Chapel—the meeting place of the House of Commons—the Lords Chamber, the Painted Chamber and the official residences of the Speaker and the Clerk of the House of Commons.
The actions of Superintendent James Braidwood of the London Fire Engine Establishment ensured that Westminster Hall and a few other parts of the old Houses of Parliament survived the blaze. In 1836 a competition for designs for a new palace was won by Charles Barry.  Barry's plans, developed in collaboration with Augustus Pugin, incorporated the surviving buildings into the new complex. The competition established Gothic Revival as the predominant national architectural style and the palace has since been categorised as a UNESCO World Heritage Site, of outstanding universal value.
Augustus Pugin