Why was King David spared of the death-for-death rule of Exodus 21?

It's a difficult passage. Psalm 51 is central to the incident concluding that forgiveness is for God alone not primarily the sacrificial system of the Mosaic covenant.

Psalm 51 (NIV)

... When the prophet Nathan came to him after David had committed adultery with Bathsheba.

¹⁶ You do not delight in sacrifice, or I would bring it; you do not take pleasure in burnt offerings. ¹⁷ My sacrifice, O God, is a broken spirit;

Psalm 51 is a prayer for forgiveness for this specific incident, i.e. two violations of the Ten Commandments adultery and murder, which in grace of God was granted. Just to point out that first, David condemns himself:

2 Sam 12 (NIV)

⁵ David burned with anger against the man and said to Nathan, “As surely as the Lord lives, the man who did this must die! 6 He must pay for that lamb four times over, because he did such a thing and had no pity.”

⁷ Then Nathan said to David, “You are the man!

¹³ Nathan replied, “The Lord has taken away your sin. You are not going to die.

Verse ^13a is why he doesn't die within the context of Psalm 51.

But it's not that simple ... The next sentence 2 Sam 12 ^13b is very difficult - it appears that the punishment that should be due to David (death) is placed on the first child of David and Bathsheba. Maybe that is what the OP is really hinting at: someone might comment on the significance, but nevertheless it's still difficult and shocking, then again so is the treatment of Uriah (and so was the Crucifixion). It must be noted none of the New Testament writers allude to this incident - but they would all be aware of it; excerpt from Matt 1:

^6b David was the father of Solomon, whose mother had been Uriah’s wife,

Matthew doesn't refer to Bathsheba by name and appears to be (pretty strongly) hinting at 2 Sam 12.

The backdrop must be the Davidic covenant in 2 Samuel 7 that a descendant of David is to reign on the throne over the people of God. Why make a covenant to a guy who is just about going to be condemned?

In the first place, laws are not carried out unless there is some authority which can enforce them. Who can enforce a law against a king of that era?

In any case, who knew about it, apart from Joab and Nathan? Joab has no reason to blab, esecially since the potential for blackmail helps to secure his post as commander.

Why was King David spared of the death-for-death rule of Exodus 21?

If one was to give an answer as to why King David was spared from the death penalty for orchestrating the death of Uriah the Hittite would be that in the end God knew the King David would repent of his sins and thus in some measure be an example of a sinner returning to the friendship of God. For this reason, David was spared the Biblical principle of reciprocal justice measure for measure known as "an eye for an eye" commandment found in the Book of Exodus 21:23–27.

23 But if there is serious injury, you are to take life for life, 24 eye for eye, tooth for tooth, hand for hand, foot for foot, 25 burn for burn, wound for wound, bruise for bruise.

26 “An owner who hits a male or female slave in the eye and destroys it must let the slave go free to compensate for the eye. 27 And an owner who knocks out the tooth of a male or female slave must let the slave go free to compensate for the tooth. - Exodus 21:23-27

Another conundrum to this mystery why David was spared resides in the fact that as king he was also judge. “The LORD gave victory to David wherever he went. So David reigned over all Israel, and he administered justice and equity to all his people” (1 Chronicles 18:13–14). We see later the Prophet Nathan to King David pronounce judgement on himself. God thus deceived David making judgement over others through the Prophet Nathan, just as he himself had Uriah the Hittite killed in war by deceptive manner so as to marry Uriah’s wife Bathsheba; so too he was deceived by the hand of God's prophet was realized.

The LORD sent Nathan to David, and when he came to him, he said: “Tell me how you judge this case: In a certain town there were two men, one rich, the other poor. The rich man had flocks and herds in great numbers. But the poor man had nothing at all except one little ewe lamb that he had bought. He nourished her, and she grew up with him and his children. Of what little he had she ate; from his own cup she drank; in his bosom she slept; she was like a daughter to him. Now, a visitor came to the rich man, but he spared his own flocks and herds to prepare a meal for the traveler who had come to him: he took the poor man’s ewe lamb and prepared it for the one who had come to him.” David grew very angry with that man and said to Nathan: “As the LORD lives, the man who has done this deserves death! He shall make fourfold restitution for the lamb because he has done this and was unsparing.” Then Nathan said to David: “You are that man! Nathan’s Indictment. “Thus says the LORD God of Israel: I anointed you king over Israel. I delivered you from the hand of Saul. I gave you your lord’s house and your lord’s wives for your own. I gave you the house of Israel and of Judah. And if this were not enough, I could count up for you still more. Why have you despised the LORD and done what is evil in his sight? You have cut down Uriah the Hittite with the sword; his wife you took as your own, and him you killed with the sword of the Ammonites. Now, therefore, the sword shall never depart from your house, because you have despised me and have taken the wife of Uriah the Hittite to be your wife. Thus says the LORD: I will bring evil upon you out of your own house. I will take your wives before your very eyes, and will give them to your neighbor: he shall lie with your wives in broad daylight. You have acted in secret, but I will do this in the presence of all Israel, in the presence of the sun itself.” David’s Repentance. Then David said to Nathan, “I have sinned against the LORD.” Nathan answered David: “For his part, the LORD has removed your sin. You shall not die, but since you have utterly spurned the LORD by this deed, the child born to you will surely die.” Then Nathan returned to his house. The LORD struck the child that the wife of Uriah had borne to David, and it became desperately ill. David pleaded with God on behalf of the child. He kept a total fast, and spent the night lying on the ground clothed in sackcloth. The elders of his house stood beside him to get him to rise from the ground; but he would not, nor would he take food with them. On the seventh day, the child died. David’s servants were afraid to tell him that the child was dead, for they said: “When the child was alive, we spoke to him, but he would not listen to what we said. How can we tell him the child is dead? He may do some harm!” But David noticed his servants whispering among themselves and realized that the child was dead. He asked his servants, “Is the child dead?” They said, “Yes.” Rising from the ground, David washed and anointed himself, and changed his clothes. Then he went to the house of the LORD and worshiped. He returned to his own house and asked for food; they set it before him, and he ate. His servants said to him: “What is this you are doing? While the child was living, you fasted and wept and kept vigil; now that the child is dead, you rise and take food.” He replied: “While the child was living, I fasted and wept, thinking, ‘Who knows? The LORD may grant me the child’s life.’ But now he is dead. Why should I fast? Can I bring him back again? I shall go to him, but he will not return to me.” -2 Samuel 12:1-23

Critics may have harshly criticized David’s character, but God see things we do not and can bring good out of evil. Many may not have considered the difficult circumstances in which he lived or the manners of his age.

God does not take any pleasure in the death of the sinner, but prefer he repents and turns from his way and live! In this, Scriptures give us an example in King David himself.

But if the wicked will turn from all his sins that he hath committed, and keep all my statutes, and do that which is lawful and right, he shall surely live, he shall not die. - Ezekiel 18:21

Say unto them: ‘As I live, saith the Lord God, I have no pleasure in the death of the wicked, but that the wicked turn from his way and live. Turn ye, turn ye from your evil ways; for why will ye die, O house of Israel?’ - EZEKIEL 33:11

Let us also remember that King David was a repentant sinner as well as a prophet and psalmist of the Most High!

The OP question begs an additional query: by whom or what was David spared? I will attempt to address this first assuming he was spared by the Law and second assuming he was spared by God.

Spared by the Law

The Book of Judges looks at the period prior to the Israelite monarchy as a lawless one. The Law of the Torah was seldom enforced. The narrator tells us several times that "In those days there was no king in Israel; every man did what was right in his own eyes." (Judges 17:6) The coming of King David laid the foundation for a judicial system that would enforce the Torah Law, but at the time in question this had not been established. Thus, although David admitted his guilt and Nathan declared that he deserved to die, there was no judicial mechanism to carry out the sentence. As long as David held power, he was effectually above the Law.

Spared by God

The question then becomes why did God spare David? The answer must be because God loved him so much that He suspended the judgment of the Law. When God rejected Saul as king, the prophet Samuel declared "the Lord has sought out a man after his own heart." (1 Sam. 13:14) David was that man. Not since Adam had God rejoiced in such a man. God's love and commitment to David are poignantly expressed in Psalm 89:

I have set the crown upon one who is mighty,
    I have exalted one chosen from the people.
I have found David, my servant;
    with my holy oil I have anointed him;
so that my hand shall ever abide with him,
    my arm also shall strengthen him...
 My faithfulness and my steadfast love shall be with him,
    and in my name shall his horn be exalted.
I will set his hand on the sea
    and his right hand on the rivers.
He shall cry to me, ‘Thou art my Father,
    my God, and the Rock of my salvation.’
And I will make him the first-born,
    the highest of the kings of the earth.

David was in love with God, and that emotion was returned with boundless affection. So in the end, the answer must be that God loved David with such a powerful love that, upon witnessing David's grief and repentance for his sin, God's mercy for him was more powerful than his justice based on the Law.

Ex 21:12-14 is "case law", about how to administer justice when someone assaults and kills a person deliberately as contrasted with accidentally. It is implicit that the killer did it with his own hands. Ex 21:14 taken out of context can be stretched to apply to David, but I don't think it's responsible exegesis since the focus of that "case law" is afflicting bodily injury directly (person to person) which results in death. What David did was murder through the agency of war by commanding Joab to place Uriah in a precarious position. It is like hiring a hitman.

Through Nathan, God STILL charged David with murder, but the natural reading is that God charged him with violating the 6th commandment (Ex 20:13), a "general law" that has no explicit associated punishment in the Ten Commandments text (Ex 20:1-17). In fact, David violated THREE of the ten commandments, coveting another husband's wife (#10), adultery (#7), and murder (#6), a combo that unfortunately we see too often even in our times.

But who would expose him? The only witness was Joab but then it was not in his interest to "turn David in", since Joab was shown to be very political in other stories. Thankfully, God, who sees all things under the sun, needed no witness and STILL administered the justice presupposed by both "general law" and "case law", consistent with his character that is BOTH just and merciful. Through his prophet Nathan, God:

• judged the act to be both "murder" and "stealing wife" (2 Sam 12:9, "... For you have murdered Uriah the Hittite ... and stolen his wife.") and

• punished David with punishment fit for the crime and for a king as Nathan prophesied in 2 Sam 12:10-12:

  ¹⁰ From this time on, your family will live by the sword because you have despised me by taking Uriah’s wife to be your own. ¹¹ “This is what the Lord says: Because of what you have done, I will cause your own household to rebel against you. I will give your wives to another man before your very eyes, and he will go to bed with them in public view. ¹² You did it secretly, but I will make this happen to you openly in the sight of all Israel.”

In the narrative, the prophecy was shown to be valid through a SIGN (like in some other prophecies) where 7 days later, the "fruit" of the adultery (2 Sam 12:18) died as predicted (2 Sam 12:14). And as how God treated other repentant kings of Israel / Judah, God gave some mercy which in this case sparing David's life (2 Sam 12:13b).

But notice how even David's repentance didn't prevent the prophesied punishment from becoming fulfilled with the rebellion of Absalom (2 Sam 15-16) and the giving of his concubines to another in a public humiliation (2 Sam 16:22). So even though God protected David's life until his natural death, we see that peace was withdrawn from David's family as early as 2 Sam 13 where David's beloved first son Ammon died after he raped his half brother's sister, Tamar.

(Some material was obtained from How did God discipline David and Bathsheba).

Conclusion

Consistent with the PURPOSE of commandments and punishments revealed more fully later in the NT, commandments and punishments are PEDAGOGUE (teacher, Gal 3:24-26) so we are transformed to be more like God who is JUST and MERCIFUL. The story clearly shows how through God's providence, God taught David BOTH justice by having David experienced death of his loved ones (the child, Ammon, Absalom, etc.) as well as the public consequences of adultery AND mercy by sparing David's life.

As readers and observers, we are left satisfied to see God's wisdom in orchestrating events so that not only justice is served by God's avenging on behalf of Uriah but that David learned his lessons for his own sanctification (cf Ps 51) as well as seeing that ultimately "no one is above the law".