A few things:
The central issue of this case is that it's NOT a choice between his life or the life of you or your family. By definition, if the dude's not a threat anymore, that dichotomy is no longer relevant.
The law would punish you in this situation not because you've violated anyone's rights, but because you've voluntarily chosen to murder someone who is not a threat to you. So the issue is not whether or not you've chosen to value your family over the rights of an intruder, it's whether or not you've murdered someone, and whether or not you had a valid justification for doing so.
If you're seriously suggesting that it's reasonable to execute an unconscious person because he might wake up, you probably ought to think that through a little bit more. There are any number of things besides murder that would be equally effective in preventing an unconscious person from becoming a threat again (e.g. tying them up, locking them in a closet 'till the cops show up, etc). I'm getting the distinct impression that you're not interested in killing such a person because you actually believe he'd seriously still be a threat at that point, but because you think it's just to kill such a person under any circumstances (please correct me if I'm wrong). This is, in my opinion, an extremely dangerously anti-social (or anti-societal, more precisely) stance to take.
If you're on a jury, you have a legal obligation to follow the instructions given to you. Since you seem unwilling to do so, my advice to you if you're ever up for jury duty is to tell the attorneys exactly what you just said (that you'd never vote guilty) because that would more or less guarantee that the prosecutor will kick you off the jury.