I recently purchased a 12,000 btu window air conditioning for a 400 square foot room. The unit works great, until the late afternoon around 4pm, where it struggles to bring down the temperature to a comfortable level. The room has a very large west facing window which gets a lot of solar gain, so I decided to install an outdoor roller shade. This decreased the heat coming off that wall/window significantly, but didn't really help the a/c unit out in the later afternoon at all. I checked the doors and windows, and there doesn't seem to be any gaps where heat could be coming in, what is the culprit here? Should I install another unit to help cool down the room?
Get an infrared thermometer.
With this you can measure the temperature of your walls and ceiling, and you will find where the heat seeping into the room is coming from.
I also have a large window facing south, which turned the living room into an oven in summer. I installed a vegetal awning like this:
It makes a huge difference, it looks nice, and the leaves fall off in winter, letting the sun through.