Gear cables fray from metal fatigue. The drum on levers has a relatively small diameter, and as the cable winds on and off while shifting the repeated bending eventually gets to it, leading to fraying and breaking.

I can't give you a life expectancy in miles or years because it isn't related to time and distance. It's a function of how often you shift. Folks riding the roller coaster terrain of places like Connecticut will have shorter cable life than those in the plains or even in the mountains of Colorado, who can ride many miles without shifting.

There are lots of posts on extracting the broken end of the cable from DA levers, so do a search and see if you find somehing that works for you.

For future reference, cables usually give you a bit of warning before failing completely. The lever action becomes stiffer (as it did for you) a short while before the cables let go. When you feel the change, finish your ride and replace the wire at your first opportunity, before it becomes harder to remove.