The type of injury can make a difference -- such as is the injury due to disease, medication toxicity or direct injury.
As long as the injury to the kidney is not too extreme or extensive, the kidney will recover. The kidneys have significant reserve function, so they can sustain some injury without risk. Also many individuals live with one kidney, which is all you really need.
In the extreme situation when the kidneys no longer function adequately, dialysis is used to replace kidney function, either long term or until a new kidney is provided (called a kidney transplant).