My sacrifice, O God, is a broken spirit; a broken and contrite heart you, God, will not despise. Proverbs 51:17

Commonly a broken heart is associated with disappointment in love. But that is not what is mentioned in the Scripture here.

It is the broken heart/spirit that is grieved under a sense of sin; has true repentance for it; is smitten, wounded, and broken with it, by the work of the Holy Spirit.

Clearly King David's spirit was broken when he understood the enormity of his sin against God. His genuine brokenness led him to repentance. He poured out his heart's plea at His feet and received God's forgiveness.

His heart did not remain broken. It was mended by the love of God. We know of this from the very many happy psalms King David has written/sang.

Same way, when we reach out to God with our broken heart, offering a true repentance, He will reach down and calm us immensely with His love.

The LORD is close to the brokenhearted and saves those who are crushed in spirit. Psalm 34:18

He heals the brokenhearted and binds up their wounds. Psalm 147:3