Please. It's not a "traffic nightmare." That is a gross overstatement by people who don't live in Los Angeles. I live here - traffic is heavy, but very much in line with any other major city.
Stay at the Rennaissance in Hollywood which is in the middle of Hollywood and convienent to the train, across the street from the El Capitan (which plays first run Disney movies and rotates old Disney movies as well. For example, Peter Pan had a two week engagement in February) and next door to Grauman's Chinese.
I'd suggest spending late Monday at Universal Studios and doing the studio tour, which alone is worth the price admission (don't think of Universal Studios as a theme park - think of it as a movie studio tour with a bunch of rides as extras). Find the pamphelts around your hotel, they usually have coupons for $10-$18 off admission in them. You can take the subway to Universal Studios, it is the next stop from Hollywood Highland and takes a grand total of 4 minutes. $1.50/person, no traffic, no parking expense. Then that night you can grab dinner at Citywalk or in Hollywood itself; I would suggest Umami Burger (walking distance from the hotel, high quality food at good prices and an LA burger institution) if you want somewhere trendy, or venture to nearby Thai Town in East Hollywood (you can take the subway, but for a non-local it can appear to be a little sketchy, so I'd take a cab) and have the best Thai food outside of Thailand. Also in Hollywood, lot's of touristy museums like the Hollywood Wax Museum that are open very late, even on weekdays, so plenty to do at night.
Then Tuesday go and explore Hollywood walking around your hotel, seeing Grauman's etc. Grab one of the van tours - they are everywhere on Hollywood Blvd. and take you around Hollywood, Hollywood Hills and Beverly Hills. Usually around 2-3 hours long, and remember, the price is almost always negotiable. Another great spot if Griffith Observatory - it's free, beautiful and awesome, and right next to Hollywood. It's best at night, but unfortunately closed on Mondays.
You can just take a cab to/from Hollywood, but if you prefer going the budget way, take the Flyaway from LAX to Union Station ($7/person; approximately 25 minutes in light traffic, 45 minutes in heavy traffic as it takes a fairly uncongested route) and then connect to the subway system, which is 20 minute ride from Union to Hollywood/Highland (the Rennaissance is attached to the station).
