You ask about Hashem's forgiveness. Bear with me please...
OK. Let's say it was your wife who you were unfaithful to, instead of Hashem, OK? How would you make a 'proper apology' to her? You did what she feared most. You hurt her. She is shaken. What can you really do today to say "I'm sorry" to her in a way that will mean something to her, and how can you begin to make up for what you did?
What does marital infidelity have to do with G-d? Let me try to explain what I'm getting at...
We are all unfaithful to Hashem (we are not on GYE to renew our Boy Scout membership). Hashem certainly takes our unfaithfulness personally. Though it is obviously only for our sake, the Torah does talk openly about Hashem being "jealous". Rashi understands the "rei'acha" in Mishlei ("rei'achav'reya avicha...") to refer to specifically to Hashem, and R' Akiva points out that rei'acha (in "v'ohavto l'rei'achakamocha") refers specifically to your wife. These are not coincidences.
A long time ago, I tried to post about how the natural relationships Hashem gives in the course of life are a source of normal development for a yid, until we are eventually ready to be truly alone with Him. Our relationship with Hashem is manifest symbolically at each stage of that relationship. The relationship with a spouse can become the closest one by far, and can continue to develop way past the parent/children relationships (which teach us selfless giving without payment) into elder life. And as with all the other stages given to us, as it develops, we find ourselves developing.
The relationship between husband and wife is not mainly about giving life and sustenance (as it is with our children), nor mainly for fearing, loving, and tending (as it is with our parents). Rather, it is designed to be the relationship - connecting for connection's sake, because you belong together. Ein ish mais ella l'ishto, v'ein isha meisa ella l'ba'aloh'. They are to be the "rey'im ha'ahuvim... b'gan Eiden mikedem". They are the tikkun for Odom and Chava, split apart in order to be brought together "v'dovak be'ishto, v'hayu l'bosor echad".
But we all know that Messilas Yeshorim tells us that the entire object of life and the Torah is d'veikus baShem!
Well, it is. And that's why he designed "v'dovak b'ishto" into the plans. It's all because we need to learn how to attach ourselves - how to be intimate with whom we belong. And we do not grow by reading or thinking. We only truly learn by doing. Life is supposed to give us everything we need for the rest of the trip.
So how would you meaningfully apologize to Hashem? I say it is by honoring the relationships in your life. Learn how to be useful to whoever He put into your life, with gusto. Make it a priority to be of service. And put yourself in His care by really starting to depend on Him. Start trusting that He really knows what is best for you by doing His Will even though it seems like you are losing something.
That's all He wants. He has less interest in our tears and sacrifices, but deeply desires our trust and allegiance from this moment forward. As he told us about 2,200 years ago,"With what shall I come before G-d... with burnt offerings? Shall G-d be pleased with thousands of rams, with riv'vos rivers of oil? Shall I give up my first-born for my transgression? The fruit of my body for the sin of my soul?" ... "He has told you, Adam, what is good and what G-d seeks from you: to do justly, love mercy, and walk humbly with your G-d." (Micha 5)
How long will we frum yidden with lust problems continue to lust after "tikkun"? Till our opportunity to actually do our avodah is long over?! I say be done with it. Leave the "teshuvah lust" to the people who are really just doing the occasional aveiros. Some of you may be in that category. But if you find yourself in a rut; see that you have spent years of your life preoccupied with this struggle; and are plagued with the pain of wondering "When will this end? Surely I'll take it to my grave;" then I say to you: give up! Give up on making up for your bad choices, on fixing and winning. You lost! Fixing is not your business if you cannot even quit! Learn how to give your life to Hashem and rest in His arms. Get out of your bad habit and then work the Steps or do whatever you believe you must do to live differently so that you will not have the pain that makes your life unbearable without the pacifier of lust, porn, and fantasy.
Join with others doing the same thing. You are not alone.
PS. My wife doesn't need me to make up for anything, either. She needs me to be present and to be the decent man she always hoped to be married to. As long as I am that, she is happy. And time does heal a lot, too. The time for a verbal amends will come. But waiting for it puts life on hold. And that'd be a real shame, cuz life is good.