Giteau didn’t kill Garfield. He just put a bullet hole in him. It was the doctor’s lack of sterilized practice that killed him. They stuck too many dirty fingers in the wound until it became septic. It was the septicemia that killed him.
The doctors also tried to use a metal detector to find the bullet, but declared the machine useless as it kept beeping at everything. They forgot he was on a bed with a metal frame. So they kept probing blindly trying to find the bullet.