LAX Tower Controller reported clearing an aircraft for takeoff when the ASDE alarm went off to signal there was another aircraft on the runway; which turned out to be due to the other aircraft missing the hold line. The first aircraft took off without problem; but the reporter noted the design of the new taxiways has been causing incursions to be a recurring issue.

Narrative

Aircraft X was cleared to land on Runway 24R. As they touched down; I cleared Aircraft Y for takeoff on Runway 24L. The departure started rolling; I issued some other instructions to other aircraft; and then Aircraft X was approaching the runway exit so I told them to hold short of Runway 24L at Taxiway Victor; which they read back correctly. I observed them exit and slow down while Aircraft Y was gaining speed and rolling past about the first 1/3 of the runway.I went on to scan elsewhere when the ASDE alarm went off and said 'warning; Runway 24L occupied.' I looked at the ASDE first and saw it emphasizing the Aircraft Y that was departing and the Aircraft X; who appeared to have their nose on the black part of the runway. Seeing that Aircraft Y was likely well past 80 - 100 knots; I dismissed the idea of cancelling their takeoff. I looked out the window and visually confirmed that the Aircraft X was completely stopped. I called out 'Aircraft X?!' Hesitant to issue any instructions to them at this point; given the dire situation. I truly felt like there was nothing that could be done except watch. They responded; 'Yeah; we're stopped;' and then a few seconds later said; 'We missed the line.' Aircraft Y passed by them without further incident; I believe within 200 feet or so.This is one of; I don't know exactly how many; but too many of the same event in the same spot. Another one happened to me again 2 days later. My Manager said she spoke to the pilot of Aircraft X shortly after this incursion; who described the taxiway/runway exit as nonstandard. He said the hold bars kind of come out of nowhere because of the way that the exit is kind of gradual at first; and then a sharp turn. This is a dangerous construction that has resulted in far too many close calls. There is talk about sending a letter to airmen to use extra caution with these exits and labeling them as a hot spot on the next cycle of charts. These are great ideas; but clearly the issue is the construction of the taxiways. These are new exits built last year; and I don't believe we had this much of a problem with the way the original exits were built. If I'm being completely honest; these taxiways need to be redesigned and rebuilt. How much risk are we willing to accept? I said shortly after this incident; it's only a matter of time before a collision happens at that spot.