It's a full life, if a trifle banal

I think having it down to luck was unsatisfactory, though it might have been improved slightly if the murderer's luck had run out, Sherlock worked out enough to get them there just in time to find him dying, and they arrested the lucky not-dead person as the serial killer. And then, of course, Sherlock would have to work out and prove she/he was telling the truth, but with the real killer already dead and no longer posing a threat that might have been a bit anticlimactic.