The word "brokenness" never appears in the Scriptures.
Yet we know by inference that it is one of the most important concepts to the Christian life.
It's expressed in such passages as 2 Corinthians 12:9...
..."My grace is sufficient for you, for My strength is made perfect in weakness." Therefore most gladly I will rather boast in my infirmities, that the power of Christ may rest upon me.
or Luke 22:42...
...not My will, but Yours, be done.
Brokenness is that state in which we recognize our own weakness and limitations before God, and rest in Him, rely on Him, lean on Him, and surrender to His will.
Don't miss this: "Surrender" is a big part of brokenness.