Security is on the 3rd floor at LAX. The KE lounge is on the 4th floor. The *A lounge is on the 5th floor. As you finish checking in, I'd recommend heading to the security zone on your left as you face the rear of the check-in hall. You can use either the security on your left or your right, but the lounge is located as you exit the left security area, so it'll be more convenient unless there's a huge difference in security lines. Normally though the lines on the left side seem to be shorter if you're departing on a night flight. For day flights, I have no idea as I've never compared the two.

Just went through security 30 minutes ago (22:00) at LAX and it was the fastest I've seen it in many years. Took me all of 5 minutes to pass through and into the lounge, with only about 3 people in front of me in the priority lane waiting for the passport check, and 2 people in front of me at the scanners.