The LED light on the back of our smartphones double up as a torch whenever we need to find our way around in the dark, especially places like theaters. However, some of the phones do not have very bright LED light output, most probably to ensure the high illumination should not burn out images when taken in pitch dark conditions. With the Sony Xperia Z1, the brightness of the LED is definitely not very bright but thankfully if you have a rooted device, you can do something about that.
In case, you are interested in increasing the brightness of the LED of your Sony Xperia Z1, you can simply head over to XDA- Developers forum and download a flashable zip that would enhance the brightness of the LED. The enhanced LED works when you turn the torch on using any torch app. Once you flash the zip file, you will feel the difference only in the torch application though, not in the general LED flash when the flash is turned on while taking pictures.
However, as a word of caution, make sure you turn the flashlight out quickly, because with the enhanced brightness, the phone does start to heat up pretty quickly and that can potentially damage the LED as well as the internals of the smartphone. You must go through the forum thread linked above to download the zip and read through more details in case you may have them.