I have an overnight layover at LAX and want a luxury hotel nearby for a comfortable stay. Preferably something with a free airport shuttle or easy access to private car services.

I live near LAX.

There are lots of chain hotels close to LAX, but if you look at ratings on booking dot com, you'll see that none of them get exceptional reviews.

Two hotels to look at in nearby Manhattan Beach:

Westdrift (Autograph Collection)
Shade

Westdrift gets better reviews, but Shade is in a better location, especially for dining. You'd have to investigate the airport shuttle thing, but Uber/Lyft and taxis are very available in MB.

The Ayers Hotel very nice but they have no shuttle service to the airport.(They are in an industrial area near the freeway so easy to leave in the morning if you decide to rent a car and drive there). If you want curbside Uber, you have to use Uber Black. Otherwise you have to take a bus to the LAX-it lot.

Some of the El Segundo hotels have shuttles but you will want to very the hours. Per hotels.com, Fairfield Inn has a 24 hour shuttle.Not luxury by any means though.

There is the Belamar in Manhattan Beach, also, apparently HIlton has taken it over so check recent reviews.

Just FYI re the Ayers Hotel, it's not just "near" the freeway, one side of the hotel is within about 30 yards of lanes of the 405 Freeway. On a second side is a freeway on/off ramp. A third side faces a McDonald's and huge/busy parking lot. The fourth side overlooks a gas station. The hotel may be very nice inside, but it's in a god-awful location that screams anything but "luxury."

I mentioned that it was in an industrial area. I have been inside, but only the ground floor courtyard, the restaurant and bar.The design is around a courtyard, and I'm assuming that it was built with good windows as I don't see any comments about noise intrusion (I don't think either of us have personal experience with the rooms). Excellent ratings on Expedia and TripAdvisor. OP did not indicate she wanted anything more than a night in a 5* hotel after a flight. If that is all she wants, then I would not recommend a hotel in the middle of the City of Manhattan Beach, with the confusing circulation pattern, extra time and distance from the airport and the following morning, especially at twice the price (for the Shade Hotel).
Many people find the Ayres convenient as an airport hotel.

BTW I misspelled it , the name is "Ayres" not Ayers. It's a small Southern California chain.

The OP asked for recommendations of a "luxury" hotel and did not mention budget. My two suggestions (Shade and Westdrift) are both closer to LAX than the Ayres, and traffic congestion in the area around the Ayres is worse than anywhere in Manhattan Beach because of the freeway in/off ramps, fast-food restaurants and nearby Costco.

If proximity to LAX is less important than the level of luxury, the Ritz-Carlton in Marina del Rey is about the same distance from LAX as the Ayres. The Ritz-Carlton is the only 5-star hotel in the LAX area.

There are not significant distances between LAX and any of the various hotels mentioned here, as Google Maps will show. However, if the OP is driving, the confusion of one-way streets near the Shade Hotel is definitely something to consider. In my experience getting from anywhere to Marina del Rey can be extremely problematic. If the OP is heading south after her potential overnight, that is the last place she should stay. I>M>O

I don’t think the OP was all that interested. He started three threads in quick succession and hasn’t bothered to return

Yep... Just California dreamin'?