It is an absolute impossibility to 'repent' or even have remorse for something unavoidable.
You have defeated your own argument. David was confessing his sinfulness in a psalm of repentance. He was confessing that he was a wicked sinner.
Paul often did the same thing. Consider the statements:
I am the chief of all sinners.
O wretched man that I am; who shall deliver me from the body of this death!
It is not I but sin that dwelleth in me.
In the last statement Paul points right to his sin nature. The sin nature that dwells within him causes him to sin. Think about it. Does the act of sin dwell in a person. No. The nature to sin dwells in a person. The act of sin (murder or lying) is done because of the sin nature that is within the person. The act is done without, that is, outside of the body, not within the body. It is done because sin or the sin nature lies within the body. It dwells or has taken up residence there, and has been there since birth. That is what Paul has testified to.
That is what David confessed to.
If you have never confessed to the Lord that you are a sinner, isn't it time?