I worked around the problem on my Nexus5 which had gotten wet and the Touch Screen function stopped working. Screen was not damaged, but horizontal lines on lower part makes it hard to read some things unless phone is rotated.
I purchased a micro USB OTG (On-the-go) cable from Fry's electronics for 4.99 and connected it to my phone USB/charging port with a old computer USB mouse connected. I was able to use it to Unlock my phone using the mouse, and go thru settings to disable the screen lock, then launch my apps to export my important files, and to then use USB cable and email file transfers to save the data.
I will next try to repair the Nexus5 phone by replacing the main board, and sell it used. I had already replaced my phone with a Nexus 5X, but was so unhappy with the poor memory and other problems of the 5X, I ordered 6.
Note: When the USB mouse is connected, battery usage is high, and there doesn't appear to be an easy way to charge and use the USB mouse at the same time.