I have a few black vertical lines on the side of my iPhone 5s screen.

A screen is composed out of different components. On the inside you have the LCD which displays the actual images, in the middle you have the 'digitizer' which responds when you touch the screen, on top you have the protective glass.

When discoloration or vertical / horizontal lines occur, the LCD is most likely cracked or damaged by the fall; even though no cracks might be visible.

You should try to replace your screen, it's fairly easy and iFixit has a guide: iPhone 5s LCD及数字转换器更换教程