Answer: Based on your description I am not certain how much light the plants actually receive. Impatiens should have done well in a low light situation, so perhaps there are more contributing factors than just light. In general, afternoon sun is very hot sun and you would need to select plants that can tolerate that for a west facing windowbox. If the sun is brief but hot (meaning an hour or two between noon and about four in the afternoon) then you will need to experiment until you find something that is happy in that particular spot. It is also important to keep the soil evenly moist without allowing it to dry out or making it sopping wet. To help with that, you might try adding some water holding polymer to the soil mix. It also helps keep the soil cooler if you use a deeper and wider container.
Here are some things to try for that hour or two of hot afternoon sun: annual dianthus, dusty miller, bedding begonia, marigold, sun coleus, "spike", asparagus fern, thyme. If the box receives sun all afternoon, try heat and sun lovers such as salvia, flowering vinca rosea, zinnia, trailing verbena, and so on.
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