How about this as an academic argument:
The key phrase of the Second Amendment is found in the preamble, but not the militia part. The key phrase is "necessary to the security of a free state." Why the adjective? The flip answer is that a state that bans the possession and use of arms by the "people" is not as free as one that does not. But there is a more mechanical aspect that supports a right to self defense.
A free state implies necessary restraints on the armed agencies of government that are vested with the authority to use force. This lessens the risk that such entities will become agents of tyranny, but also impedes their ability to defend the life and safety of individual citizens. Unless the people are willing to accommodate an oppressive police presence in the interest of public safety (airline security lines notwithstanding) the citizen of a free state (i.e. one in which government force is restrained in the interest of individual liberties) must be allowed the right and means to provide their own defense.
As an aside, I think that probing whether English common law protects a right of self defense is largely unhelpful. It is my understanding (quite possibly wrong, but...) that people living in England were subjects, and that the King owned some sort of interest in his subjects' lives beyond humanitarian or humanistic ones. Killing one of the King's subjects was almost as bad as killing one of the King's cows. People living under the U.S. Constitution however are not subjects in that sense, and are not afflicted by the disabilities imposed by monarchical abstractions.
To sum up: "Free" implies limitations on the state's ability to provide an arbitrary level of security, this necessarily implies a degree of self reliance on the part of the free citizen to protect his life in the setting of such limitations. If the Constitution prevents the policeman from looking in the trunk of the speeding hit man's car when he is on the way to kill you, the Constitution impliedly allows you to rectify that by dispatching the miscreant when the choice comes down to you or him.