ITT naive fucktards who don't know how the real world works. It's called cutting your losses. It's a harsh reality of life.
Euphemia had already been accidentally forced to kill untold numbers of Japanese. At this point, killing her is charitable.
As for her reputation, it was already ruined. Something as vain and as pointless as a reputation could have been salvaged had Lelouch come out to the world about his Geass and how it happened, but his revolution would have ended at that point, and not only would everyone Euphemia killed have died in vain, but the same would have been doubly true of everyone who had died previously as a result of his now failed revolution.
Lelouch is a realist who handled a horrible situation masterfully to reach the best possible end for all parties concerned. And don't say "THAT'S NOT TRUE EUPHEMIA'S REPUTATION IS RUINED AND SHE DIED". That was already set in stone by the accident in question, which is what it was, an accident.