Hm, this question is rather set up as if breaking into someone’s house is an unforgivable sin. For all you know, this stranger breaking into your house is a Christian, and thus killing them won’t send them to Hell. Christians still knowingly sin, and that is something that one might do when very desperate.
For my part, I wouldn’t kill them. First off, because I wouldn’t have a weapon that I could do so with. My reason for not having that weapon wouldn’t be because I never expected to have someone break in, but because I wouldn’t want such a recourse at my disposal in such an event.
In this case, 911 has already been called. As soon as 911 was on the phone, I would vocally let the invader know that the police are already on their way so that the robber had better be leaving immediately. There is no need to force a confrontation, unless the invader has already come into contact with someone in my household, in which case the police would be better equipped to deal with the situation anyway. There is no reason to go out to investigate: that is the most likely way to make the situation end badly.
If for some reason I did go to the stairs and see the person down there start up them, I would let them know that I’m there, and that the police are on their way, and advise them to leave. If possible I’d do it while getting out of sight in case the stranger has a gun. It is highly unlikely at this point that the robber will not leave: they do not have the time to do any significant plundering, and they know that advancing and trying to take a hostage would not likely end well for them, certainly not as well as they’d expect running away to end for them.
Robbers, by and large, only carry on their robbery until they are detected. Most will promptly leave once they know they are found out. Immediately killing, or attacking, someone who will most likely run if you give them the chance, is completely unethical.
*Child of Feanor*