Who was to blame for Britain's failure to win a quick victory over the American rebels: General Howe, General Burgoyne, or the ministers in London?

Failure on the part of the British to win a quick victory over the American rebels can only be placed on the grounds of a retrospective, hypothetical appraisal of what would have happened had one battle or event had a different outcome. It is important to note that this thought experiment is hardly a conclusive exercise, and it is possible that the British army would have been thwarted even had each of these three parties behaved differently.
Most prominent of the ministers in London was Prime Minister of Great Britain, Lord North. North made an unusual and creative conciliatory effort toward the colonists in 1775: He offered to relieve colonies that helped to suppress rebellion of paying taxes. The Continental Congress ignored his efforts, and North had no way of carrying them out. Still, the fact that North made such a gesture (added to the fact that he was only minimally capable of affecting matters on the ground in the colonies), suggests that he is less responsible on an individual level. North also demonstrated a focused effort on tactical matters. His was the idea to isolate New England by means of a three-pronged attack against Albany, New York (using the forces of General John Burgoyne, Colonel Barry St. Leger, and General Howe in concert).
British officer John Burgoyne seems to have been more assertive and determined than General Howe. Burgoyne, a British general who fought in the Seven Years' War, was appointed major-general in the American Revolution. Burgoyne was dutiful in bringing his troops south from Quebec to Saratoga (after having saved Quebec from siege by the Continental Army). While Burgoyne might have been overconfident in his ability to defeat the rebels on this march, the failure of this move to isolate New England owed to General Howe's renegade move to campaign against Philadelphia instead of aiding Burgoyne.
Howe unilaterally decided to attack Philadelphia, where the Continental Congress regularly convened. This move was somewhat successful, as he was indeed able to march on Philadelphia); however, the Continental Congress fled and thus robbed Howe of a total victory. Moreover, the opportunity cost of Howe's strategy was significant; because he was with his troops in Philadelphia, he was not able to aid Burgoyne's troops at Saratoga. Burgoyne, assailed by militiamen from New Hampshire and Massachusetts as well as rebels from New York led by Horatio Gates, waited for help that did not come. The British commander in New York recalled 4,000 troops intended to go to Albany to support Burgoyne, in order to send those troops to Philadelphia to support General Howe. Ultimately, Howe's maverick tactic in this northern theater brought defeat to himself as well as Burgoyne.