The short answer to the question of why Lady Macbeth didn't kill King Duncan is that she thought about doing it but couldn't bring herself to go through with such a bloody deed because he looked too much like her father.
The long answer, however, is that she did not kill the King because the play's author did not want her to kill him. Shakespeare wanted Macbeth to do the killing because the play is about Macbeth, what he does to become king and how this affects him afterward. Lady Macbeth exists as a character to give Macbeth someone to talk to. Characters in plays have to talk to each other in order to communicate information to the audience. Macbeth has no one else with whom he could discuss his most secret thoughts and feelings. His wife also serves to encourage him to go ahead with his ambition to become king by murdering Duncan.
Then why does Lady Macbeth say that she would have killed Duncan herself if she hadn't thought he resembled her father? Mainly because Shakespeare wanted to explain to the audience why such a vicious woman didn't actually commit the murder when she had the opportunity. Shakespeare also wanted to explain to the audience that she "laid their daggers ready." Why was this necessary? Shakespeare probably wanted to explain how Macbeth was going to be able to get inside the King's bedchamber and assassinate him when he was being guarded by two men. Lady Macbeth's soliloquy informs the audience that she was an important accessory. She provided the grooms with drugged "possets." (The word "possets" is glossed in one edition as "hot drinks, containing milk and liquor.") Her statement that she "laid their daggers ready" is intended to inform the audience that Macbeth will have the weapons available when he enters the bedchamber and, more importantly, that he plans to kill Duncan with the grooms' daggers in order to be able to frame both of them for the King's murder.
Why would these two men want to murder the King? They were "suborned." Someone who wanted the King dead had supposedly bribed the grooms to commit the murder.
They were suborn'd:
Malcolm and Donalbain, the King's two sons,
Are stol'n away and fled, which puts upon them
Suspicion of the deed.
When Macbeth returns to his bedchamber still holding the bloody daggers, the audience will understand that these daggers belong to the two grooms and hence will understand the whole plan: Drug the grooms, kill the King in his bed, leave the bloody daggers with the grooms to frame them and cast suspicion on some unknown person or persons who had a motive for wanting the King dead. Suspicion naturally falls on Malcolm and Donalbain after they flee, but the real mastermind might have remained a mystery otherwise. Shakespeare, who is responsible for everything that happens, has the boys flee because that makes it much easier for Macbeth to get elected as the new king instead of Malcolm, the heir apparent.
In Act 2 of Macbeth, Lady Macbeth and Macbeth act on their conspiracy to kill King Duncan, trying to force the prophecy to be true. In Act 1, we learn from Lady Macbeth that she does not think Macbeth has enough ambition to act on the plot. We also know that Macbeth is having second thoughts about killing Duncan, because of the new title Duncan has given Macbeth. It is curious, then, that Lady Macbeth doesn't kill Duncan herself. The answer is in Act 2, scene 2. While waiting for Macbeth to come out of Duncan's chamber, Lady Macbeth states, "Had he not resembled My father as he slept, I had done 't." Lady Macbeth sees a resemblance between Duncan and her father, which causes her to hesitate in killing Duncan. This fact adds another layer to Lady Macbeth's character, as it is the first hint at emotion she shows.