At the outset, I would like to flag up two biblical principles that form my starting point for answering this important question.
(1) Romans 3:20 - “By the deeds of the law shall no flesh be justified in his sight: for by the law is the knowledge of sin.” Whatever answers folks come up with, there surely can be no disputing that striving to keep either all, or some, of God’s law will never contribute in any way to being justified in God’s sight?
(2) James 2:10 - to break the written law of God at one point means all of it has been broken. “For whosoever shall keep the whole law, and yet offend in one point, he is guilty of all.” Verses 11 and 12 add the strongest caution for all Christians who would strive to retain some kind of law-keeping, saying that they SHALL transgress that law, but that Christians “shall be judged by the law of liberty.”
Now I would simply like to list relevant scriptures further to those two principles, and let them speak for themselves, in light of the aforementioned principles.
Romans 6:14-18 “For sin shall not have dominion over you: for ye are
not under the law, but under grace…to whom ye yield yourselves
servants to obey, his servants ye are to whom ye obey; whether of sin
unto death, or of obedience unto righteousness? But God be thanked,
that ye were the servants of sin, but ye have obeyed FROM THE HEART
that form of DOCTRINE which was delivered you. Being then made free
from sin, ye became the servants of righteousness.”
Ephesians 2:13-18 “But now in Christ Jesus ye who sometimes were far
off are made nigh by the blood of Christ, for he is our peace, who
hath made both one, and hath broken down the middle wall of partition;
having abolished in his flesh the enmity, THE LAW OF COMMANDMENTS IN
ORDINANCES; for to make in himself of twain one new man, making peace;
And that he might reconcile both unto God in one body by the cross,
having slain the enmity thereby; And came and preached peace to you
which were afar off, and to them that were nigh. For through him we
both have access by one Spirit unto the Father.”
Colossians 1:21-22 “And you, that were sometime alienated and enemies
in your MIND by wicked works, yet now hath he reconciled in the body
of HIS FLESH through death, to present you holy and unblameable and
unreproveable in his sight.”
Colossians 2:13-14 “And you, being dead in your sins and the
uncircumcision of YOUR FLESH, hath he quickened together with him,
having forgiven you all trespasses; blotting out the handwriting of
ORDINANCES that was AGAINST US, which was contrary to us, and took it
out of the way, nailing it to the cross.”
I have capitalised words and phrases relevant to the question, and related to the two biblical principles at the outset, which I hope will form a solid foundation for building an answer upon – the answer now being self-evident from the scriptures quoted.
Hebrews 10:15-16 is a key text in the question, which I address now. It quotes from Jeremiah 31:31-33, where I note that God promises a new covenant (singular) and a new law (singular) but written upon many new hearts, and contained within many renewed minds. This strikes me as significant when James speaks of the (singular) law of liberty – James 2:10-12 quoted in full at principle No. 2.
It is the (singular) law of liberty that is written in new minds (which are now the mind of Christ – 1 Cor. 2:16) and in new hearts of Christians.
God is not dealing with a litany of individual sins with individual Christians, because the new covenant in Christ’s shed blood dealt with ALL sin, just as ALL of the old covenant law was nailed to the cross, having served its purpose and now having been taken out of our way. So, why would any Christian want to go running back to that “handwriting of ORDINANCES that was AGAINST US”, to drag it (or even some of it) forward for consideration? It is no longer in the new covenant equation.
If a Christian has not discovered this law of liberty – their lack shown by keeping on going back to the old covenant law and ordinances, as if some of it still applies, somehow, in the new covenant – then they will have fallen between two stools. A person either sits on the stool of old testament law and ordinances (the entire lot) or on the stool of the law of liberty, found only in Christ. That’s where Christians would want to be found resting on, I suppose, for no rest will be found anywhere else. It simply is not possible to try to keep those two stools tied together, so that a Christian can straddle both. That may now call for a new question!
But my simple answer to this immensely complex matter (simple though your question actually is) is that God puts his law of liberty in Christians’ hearts and minds.
EDIT - You reasonably requested that I explain what this ‘law of Liberty’ is. Consider, please:
John 1:16-17 “For of [Christ’s] fulness have all we received, and
grace upon grace. For the law was given by Moses but grace and truth
came by Jesus Christ.” Verse 14 states that when the Son of God dwelt
among men, they beheld him to be full of grace and truth. Acts 13:39
“And by [Christ] all that believe are justified from all things, from
which ye could not be justified by the law of Moses.”
The movement is away from the law of Moses, which only condemns all who break it (all of us), to being justified in a legal sense before God. That can only happen when us law-breakers receive the fullness of Christ – grace upon grace. Then we are no longer judged by the old testament law because Christ blotted out “the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to the cross” (Colossians 2:13-15).
Needless to say, a person who has not received that fullness of Christ, that grace upon grace, will not understand – let alone experience – the liberty in Christ that comes with realisation of our new freedom in Christ. Only then does a person begin to grasp the wonder and joy of having been judged according to the law of Liberty, and thus being set free - liberated. But our sins have not been swept under a heavenly carpet. They have been dealt with legally. Individual sins were suffered by Jesus Christ in the hours of darkness, bearing our sins in his body on the tree. This deals with one's own personal actions committed within and by one's own body, mind and heart. Sin, itself, that which entered into the world by serpent, woman and man and passed to all under Adam’s headship, was dealt with in the death of Christ, sin being eradicated within his - clean - humanity, he having been 'effected' sin. That is why “the gospel of Christ is the power of God unto salvation to very one that believeth… for therein is the righteousness of God revealed from faith to faith: as it is written, ‘The just shall live by faith’.” (Romans 1:16-17) Pure justice was carried out at Golgotha, so that undeserved mercy could then be the verdict on all liberated from condemnation. That is the ‘law of Liberty’.
And that is why a new question now needs to be asked. You asked “What laws will God put in Christians’ hearts and write in their minds?” My answer was, “The law of Liberty”. You did not also ask, “And what are those laws?” Now someone needs to ask, “What, exactly, is this ‘law of Liberty’ seeing as it is not the Mosaic law?”
I have begun to deal with that, as you reasonably requested. But to do the matter justice, I would have to write screeds more. Having established what law is NOT in Christian’s hearts and minds, the way is now clear to go off in a new direction, to pursue the spiritual experience of this law of Liberty, to investigate how, exactly, this applies to Christians, what was required (legally) for God to no longer judge Christians by the condemnatory law of Moses, to grasp how Christians have no case to answer (in the heavenly court). This is a huge subject of which I have only provided tantalising glimpses. It requires a fresh question, I would suggest.