Breakfast/Lunch within walking distance of Sunset & Vermont?
Sorry to hear about your friend. Sounds like you'll be needing your strength; let's get you fed!
I'd say if you wander north on either Vermont or Hillhurst, you'll be good to go. Here's some standbys:
Alcove (opens 6AM, 1929 Hillhurst)
Fred 62 (24hr, 1850 Vermont -- closest to hospital)
Home (opens 9AM, 1760 Hillhurst)
House of Pies (24hr, 1869 Vermont)
Little Dom's (opens 8AM, 2128 Hillhurst -- furthest from hospital)
Mustard Seed (opens 8AM, 1948 Hillhurst)
These are only the tip of the iceberg but should get you started. There's also some taco places worth checking out, a fresh fruit vendor on Vermont just north of Sunset, and a couple popular coffee chains on Vermont between Sunset & Hollywood should you require that sort of thing.
I like Little Dom's but it would be a mile walk each way. Mustard Seed and Alcove are only a few blocks closer. Parking isn't hard in near either so they are good options if the OP would want to stop there prior to going to Kaiser but the distance is probably too much for a quick convenient bite unless the OP were to stop on the way.
Home, House of Pies and Fred62 are closer, albeit not as good. I'd add two more on Vermont that are closer and that I like a lot:
Le Figaro Bistrot (open for Breakfast, Lunch and Dinner)
Rocco's (a very good Italian deli)
Square One mentioned below is a great option.
Hope your friend feels better.
One good thing about that Kaiser location is that you are right on the Red Line, there is a Vermont/Sunset station. If you hop on the train, go just four stops(towards Union Station) to the Westlake/MacArthur station and you'll find Langer's Deli, home of arguably the city's top pastrami sandwich. That ride on the Red Line probably takes less than 10 minutes. Langer's does take-out, too. If you called in your order and took your food to go, you could be there and back in like 30 minutes.
EDIT: Shoot, I apologize -- you wrote within "walking distance." Clearly, Langer's is not within walking distance. Somehow I interpreted "walking distance" as "I don't want to move my car" so I included the Red Line trip. But, obviously that's "walk to the train, walk to the restaurant, walk back to the train distance."
I second Fred62 and Pie& Burger, but your nearest edible option is Square One, which does breakfast/lunch reasonably well and is located on Fountain one long block directly south of Kaiser.
For lunches, you should also check out The Kitchen on Fountain at Sunset, the Unami Burger nearby on Hollywood Blvd, and Vim Thai on Hollywood at Normandy. Good luck.
I'd avoid Home and House of Pies, as both are below average, with food that tastes like it's been pulled from the deep freeze and popped in the microwave - probably because it is.
Spitz is a new kabob place - middle eastern via Germany(?) on Hillhurst just north of Hollywood.
Mustard Seed has a great turkey burger.
Square One has arguably the best breakfast in the city.
Boo's Philly Cheese Steaks is a short walk from your location.
Alcove - while the food isn't fantastic it's certainly serviceable and the patio seating makes for a fine chill-out spot.
It's not much to look at, but Los Burritos right there on Sunset just west of Edgemont is pretty good for tacos, burritos and (my favorite) the grilled chicken platter. I'm pretty sure they serve breakfast, too. They also have a juice bar serving up fresh-squeezed juices & smoothies.
re: mucho gordo
Just wanted to thank everyone for the suggestions.
Friend finished chemo and radiation (concurrently) a few weeks ago. The treatment session was so brief, we (or me alone) never got any further than Pollo Loco --the rice bowl was something she could tolerate while on Chemo. Drive through, then race up the 101 to the 170 & home.
Good news, the treatment worked! After surgery in a few weeks to remove the now shrunken tumor, we will be exploring in the area.
Grateful for your help, Chowhounds!