Document isn't so much a coffee shop that doubles as an art gallery as it is an art gallery that doubles as a coffee shop. Not only are there rotating exhibits on the walls, it seems as though everything in the space itself is an art piece: the coffee brewed in Pyrex pitchers, the light fixtures housed in buckets, the chairs similar to the ones you sat in back in kindergarten. Which all makes sense, as the shop is owned by two artists who also use their shop as an art gallery space. It's often as quiet as an art gallery might be, making it perhaps an ideal space for your cold brew and your laptop.
3850 Wilshire Blvd. #107 | Coffee: Rotating; recently: Supersonic Coffee Roasters, Temple Coffee Roasters
To eat: Pastries from Bread Lounge
To eat nearby: Pretty much all of Ktown; try Myung Dong Kyoja for dumplings

Right in the middle of Koreatown is Balcony, a shop so serene it's hard not to sort of zen out while you're here. The flowers and greenery, anyway, are a very welcome respite from all that traffic outside; there are a few tables in here, plus a few in the patio outside. The short menu includes nitro cold brew and a signature iced drink called the Brad Pitt ^__^ And for those times you can't stand to look at your phone any longer, check out the bookshelf: Among other books, you'll find the Hunger Games trilogy plus a rather impressive collection of John Grisham's greatest hits.
314 N. Western Ave. | Coffee: Stumptown Coffee Roasters
To eat: Pastries from Sweet Lady Jane

The shop formerly known as Bourbon Street Cafe flipped over to Alchemist Coffee Project a few years ago, and while the decor's changed a bit since then (slightly more industrial, slightly more whimsical)  the coffee is as good now as it was then. Fans of Bourbon Street Cafe's New Orleans-style cold brew will be happy to know that the Alchemist team kept it on the menu; there are also lattes and shakes and "Duo," which combines espresso and milk tea (!).
698 S. Vermont Ave. | Coffee: Rotating; recently: 49th Parallel Coffee Roasters, Heart Coffee Roasters
To eat: Pastries and treats from Jin Patisserie
To eat nearby: Bossam specialist Kobawoo is in the same plaza

If you light up anytime you see "floral" listed as a flavor note on a coffee bag, you will probably very much like Bia. The shop uses all natural flower syrups (rose, lavendar) to flavor lattes, teas, and sparkling "ades," and they're all so pretty and Instagrammable that there's a sign that asks you not to forget to stir your drink after taking a photo. Priorities, right?
3907 W. 6th St. | Coffee: Klatch Coffee
To eat: Cake!

A sweet, quiet shop across the street from Dan Sung Sa. All sorts of lattes here — vanilla, caramel, honey, honey lavendar, etc — plus fun mlikshakes, too, like one with mixed grains (the misugaru).
3300 W. 6th St. | Coffee: Beau Bar
To eat: Pastries and sandwiches

The fact that everything here is served in thematically-appropriate vessels — beakers, flasks, vials — would be kitschy if not for the fact that the shop isn't very serious about it. They are, rather, having a bit of fun with it, which is a nice change of pace; as they roast their own coffee after all, there is quite a range of coffees on the menu; if you want to a good representation of what they do, try the "Mad for Caffeine," which would be a coffee flight of sorts, with two different coffees and an Americano to mix things up.  
429 N. Western Avenue | Coffee: LaB Coffee Roasters
To eat: Farmshop pastries
To eat nearby: In the same plaza as the shop is Western Doma Noodle.

caFe at POT
In the lobby of The Line Hotel in the heart of Koreatown is Roy Choi's coffee stand, a coffee corollary of sorts to POT, his hot pot restaurant that sits directly across from the cafe. Here there's a mix of old and new, low and high: the La Marzocco is pulling LAMILL shots for drinks like misuguru lattes and there's"da bang coffee"; the pastry case is one of the most expensive out there, and it's housing treats like red bean buns, almond puffs and French toast pizza. It's bright, it could be loud, it's definitely not like any other coffee shop out there. Thank the gods.
3515 Wilshire Boulevard | Coffee: LAMILL
To eat: The caFe has a fantastic congee. Also, pizza at POT on Fridays and Saturdays starting at 6pm!