Ref: Memento Explanation
Lenny believes in two things:
1) When he kills his wife's murderer he will somehow remember that he's avenged her death.
"Somehow it will be different"
2) It doesn't matter to him if he remembers or not. He feels his wife's death has to be avenged for her sake. For her justice. This is his purpose.
When Lenny is in the process of killing Jimmy, Jimmy whispers "Sammy" just before dying.
Lenny drops Jimmy surprised that he knew about Sammy. Lenny suspects
that if Jimmy knows about Sammy, then it’s someone who Lenny might
have met in a completely different circumstance. That Jimmy may have
nothing to do with his wife’s murder.
At this point, Lenny realizes that he's killed some guy who has nothing to do with his wife's murder. When Teddy comes there he roughs him up and asks for answers. Teddy tells him that a year ago he already avenged his wife's death. That even Teddy thought Lenny would remember this moment. But Lenny forgot.
Both things that Lenny believes in is invalidated.
1) He didn't remember the revenge.
2) His wife has been avenged and he has no purpose now.
However, he's angry that Teddy has been using him for a year to get rid of people. He's not in a position to trust Teddy with the information he has just received. Lenny has two choices:
1) Believe Teddy and write all of that down but wake up with no purpose left in life. Perhaps this has already happened.
Teddy also accuses of Lenny pulling out 12 pages from a full police
report that he’d given him. He says that Lenny removed 12 pages to
create a puzzle that he couldn’t solve.
2) Discard what Teddy had to say and continue looking for a John G.
Lenny feels that if Teddy thinks it’s all only about puzzle solving,
another John G to look for, then Teddy is a John G. He thinks Teddy
can be his John G. He thinks “Do I lie to myself to be happy? In your
case, Teddy, Yes, I will”.
Why doesn't he kill Teddy right then and there? At that point he knows that Teddy is not his wife's killer. He knows, just like Jimmy, he will not feel the revenge if he kills Teddy right there. He needs to lie to himself to let circumstantial evidence lead to Teddy as his wife's murderer. Easiest evidence that will lead to Teddy is his License Plate Number.
This is the climax of the film. The irony is that Teddy himself gets
marked as a false clue. A false clue that Lenny leaves himself. This
moment here leads up to Teddy getting shot.
Does this mean that once he's killed Teddy he's going to stop? Perhaps not, because he's still going to forget the feeling of the revenge that he has taken. He's going feel the lack of purpose in his life. He might just keep going on like this, creating puzzles that he can't solve.