Yeah, no KLM lounge at LAX. They use the KAL lounge (although biz gets the nice one world lounge) for economy and other STE+ travelers. It's a PP lounge, its huge so doubt they would deny entry. Let's just say this, the recent KLM economy flight, I started at T2 to use the Delta SkyClub there to avoid the KAL lounge. It's space is ok, but the food is very lackluster (best thing is the Korean instant Ramen, that I personally buy, but it's $1.50) and the booze options are very limited.

I'd buy Delta SC in T2, particularly since you are there already, it's $29 if you have a Delta AMEX, but I think $50 for a day pass. Though for the price, I'd just eat in the terminal unless it is a long layover. Or the VS lounge if you happen to be there during the morning PP hours. Great intimate space (I've been during the normal VS hours twice) but it's more basic food and beverage (unlike the made to order multiple course meals with Champagne and designer cocktails during the VS Clubhouse service).

Heading to the local beaches for food is a great option. Quick Lyft/cab ride and way better. Manhattan/Venice/Hermosa. Each are very different but cool in it's own way.