In episode 3 of Code Geass: Lelouch of the Rebellion, Lelouch killed
Clovis as he was leading operations against the resistance, and Cornelia did the same thing.
However, Lelouch did not try to kill her despite this. Why? It was even made clear that he had nothing to do with death of Lelouch's mother.
Clovis was killing innocent Elevens, not just the resistance/terrorists.
…just as former Viceroy Clovis was punished for ordering the slaughter of countless unarmed Elevens. We cannot stand by and allow such cruelty to be carried out, and so we made him pay for his actions. I will not repudiate battle on a fair and level field, but neither will I tolerate a one-sided massacre of the weak by the strong. The only ones who should kill are those who are prepared to be killed! Wherever oppressors abuse their power by attacking those who are powerless, we shall appear again, no matter how mighty or formidable our foe may be!
Source: Wikiquote - Code Geass - The Black Knights (Second Section)
While yes, Cornelia can be brutal like using a Knightmare to gun down the JLF who really stood no chance against one, she didn't order innocent people to be killed. Both she and Lelouch believe in the principle that "the only ones who should kill are those who are prepared to be killed" and fight on the front lines with her soldiers, even telling Guilford off when he tells her to keep back.
During the Battle of Narita, Cornelia leads the attack on the Black Knights along with Dalton and Guilford. Guilford warns her to stay away and she responds by destroying several Burais.
Source: Cornelia li Britannia - Character History - First Season (5th Paragraph)
Clovis, on the other hand, ordered innocent people to be slaughtered all the while he hid in his G-1, and even started to beg Lelouch to spare him. Cornelia would more than likely not beg for her life to be spared, if she was to beg I would dare say it would be to spare her soldiers in exchange for her life
Also, Cornelia was originally assigned to Marianne's Guard before being ordered by Marianne herself to dismiss the guard and the palace's defenses on the day of her death. Under Lelouch's Geass, Cornelia revealed she was investigating Marianne's death herself.
Finally, Lelouch didn't have the time to finish her off anyway. Remember that while he was interrogating Cornelia, C.C tells Lelouch that Nunnally is in danger (due to V.V having kidnapped her). Lelouch's first priority is always Nunnally as, during the first attack on the Tokyo Settlement, he had the Black Knights protect the school because she was there. During the second attack, he had Rolo and Sayoko get Nunnally out of Tokyo and broke down when Nunnally was thought to be killed. With Nunnally in danger, he abandons the attack on Tokyo and his own vendetta against Britannia so V.V could very have well saved Cornelia inadvertently.
Lelouch needed her alive because she had info on the death of his mother which could help him unravel the truth.
By the time he finally gets to her and geasses her to ask the questions, the Black Rebellion is in full swing, so killing her after the questions would be a bad idea because holding the viceroy captive is an excellent way to force a victory. Of course Jeremiah then enters the scene and everything goes haywire.
After that, Cornelia disappears until the second half of season 2.
