Steinbeck's compassion for Lenny pertaining to the part of the story when Lenny is killed because Steinbeck allowed Lenny to die at the hands of his friend. George kills Lenny in an effort to save Lenny from a possible more horrible and torturous death at the hands of the men hunting him for the murder of Curley's wife.
In the final chapter leading up to Lenny's death Steinbeck uses calm imagery describing the place where Lenny fled to to meet George. Lenny remembers Aunt Clara aloud and in these memories that he conjures he is reminding himself of how good George has been to him even though he gets into a lot of trouble. George approaches and remind Lenny of their big plans to set his mind at ease and then he squeezes the trigger, with the hopes that Lenny will go to a better place where he can tend rabbits and stay out of trouble.
Steinbeck gave Lenny a very peaceful death given the circumstances and severity of Lenny's crime and this is how he showed compassion for him.
When George kills Lennie, his dear friend and only true companion, the compassion Steinbeck demonstrates is overwhelming. George did not want to lose his friend, but he understands that a quick, clean death is much more preferrable than the brutal and torturous death Lennie would face at the hands of the angry mob. Lennie will die. That much is certain. To have the killing be as painless as possible is definitely compassionate. Before George has to pull the trigger, he makes sure he and Lennie understand one another and his love for his friend, and he gives his friend the gift of their dream one more time:
Lennie said, "I thought you was mad at me George. "
"No," said George. "I ain't mad. I never been mad, an' I ain't now. That's a thing I want ya to know."
The voices came close now. George raised the gun and listened to the voices.
Lennie begged, "Le's do it now. Le's get that place now."
"Sure, right now. I gotta. We gotta."
And as George raised the gun and steadied it, and he brought the muzzle of it close to the back of Lennie's head. The hand shook violently, but his face set and his hand steadied. He pulled the trigger.