[Los Feliz Slim]

You could drive north to Santa Barbara, but the coast drive really only starts to get good around San Simeon, around 2.5-3 hours north of LA. It's worth it, though. It should be reiterated that if we have a rainy season, this is it, and it's usually chilly at night (50's). You could drive south to San Diego and/or Mexico, if that's your thing. Or, you could go east into the desert and check out Joshua Tree. Driving east has the added bonus of Casino Morongo, which is actually very nice. They play craps with cards there.

If you drive fast, the drive to Vegas from LA takes 4-5 hours (6-7 on the way back b/c of traffic, though).

Things in LA I would recommend, some are repeats I'm sure:
The Getty Center
Sunset Strip (stay at The Standard, good place to meet people if you're alone)
The Improv
Santa Monica Pier / Promenade
The Beach Cities (Newport, Laguna, etc)
The Reagan Library (if you've gotten very bored, it was cool, but it's an hour away)
The La Brea Tarpits
Universal Studios (touristy, but I like the backlot tour)
Venice Beach

An entire week alone in one city is going to be a looong time, IMO. If I were you I would A) really try and find someone to do this with and/or B) make is a SD/LA/LV trip).