The answer to this question is twofold for the following reasons.
First, we can safely assume that Dr. Jekyll recognizes the pain and suffering that comes from a life of duplicity. He does not just see this from the point of view of society, but from a moral standpoint. In Chapter 10, almost as if giving up, he summarizes his findings on being a split individual as follows:
With every day, and from both sides of my intelligence, the moral and the intellectual, I thus drew steadily nearer to the truth, by whose partial discovery I have been doomed to such a dreadful shipwreck: that man is not truly one, but truly two.
Keep in mind that Jekyll enjoys the life of Hyde in terms of the freedom that the latter gets to experience by being, literally, a "nobody." In other words, and very true to Victorian society, the "more" you were, the more that was expected of you. In the case of Jekyll, he would have been someone so rightfully adored and respected by society that he would have always lived under the pressure of being considered "perfect" by the social standards of the time.
Hyde, on the other hand, can do as he pleases--and does what he pleases. He can be rude, excessive, compulsive, mean, violent; in fact, he can do everything that Jekyll would have wanted to do in specific situations. Therefore, it is not that Jekyll wants to be Hyde; he simply wants to be able to express himself and be free from a society that tended to suffocate its most prestigious members.
The problem came when Jekyll lost control of the transformation process and became Hyde without even trying. In a letter to Dr. Lanyon he repents this change and realizes how dangerous his situation is:
That child of Hell had nothing human; nothing lived in him but fear and hatred...
Moreover, the blended creature that Jekyll and Hyde have become apparently dislike one another. And why not? Jekyll becomes Hyde because he is unhappy with being a righteous prude. Hyde dislikes his eternal anger and ire. It is no question that, in hopes of finding a better part of himself, Jekyll only succeeded at finding the worst. Could it be that there was no redeeming persona for Jekyll in the first place, and he ended up awaking the monster within?
Therefore, there is repentance of a sort in the end when suicide seems to be the only solution. The letter is telling, and suggests the repentance comes from the standpoint of a man who tried too hard to find something, and found nothing good at all.
Here then, as I lay down the pen and proceed to seal up my confession, I bring the life of that unhappy Henry Jekyll to an end...