Welcome to SI. Sorry that you find yourself here, but know that you are in good company. Everyone here has experienced the devastation of infidelity just like you have. Sadly, there are no good or easy answers, no one can promise you an ending, and we're all just doing our best to stay positive and focused. But there are things we can do that will help us, help our spouses, and help our families, and most of those things have to do with ourselves. As you said yourself, it is the process of becoming a better person.
You and I have a few things in common. I am 56, married 25 years, have three grown children. During the affair, I stupidly brought the AP into my home, into our bed, and worst of all, introduced her to the kids and even to my wife (she was "a friend" of course). At some point, my daughter (16 at the time) noticed us being a little too affectionate, and similar to what happened with your daughter, she started to become part of the lie. I ended up treating my daughter more like a friend or an equal, and not as a parent, and in doing so, started to share our frustrations about my wife, and effectively turned my daughter against my wife. The two older kids did not find out what was going on until D-day, but it devastated them. Before my wife and I married, she had a previous husband who our two oldest are from. They divorced (the father was a gambler) and their Dad moved away and was no longer an active participant in their lives again. So for the kids, and my wife, this was a second betrayal to deal with, making things all the worse. To this day, my kids all struggle with trust issues and with committed relationships. While we've worked through a lot together, those emotional scars will never go away.
You said you are about 7 months in to the process and already seeing an IC, so that's good. You are on the right path. You should know that things from this point on will be tough. Recovery from an affair takes years, and the first year or two, at least in my experience, is just a painful time. The BS is usually overwhelmed with pain and feeling as though their entire world has been pulled out from under them and left them "kicked to the curb" with no one to trust and no real support. The WS is usually flailing, trying desperately to "make right" what cannot be fixed, and feeling buried under shame and regret, and trying to come to terms with what they did and who that makes them. For both spouses, it often feels endless and hopeless.
The best advice I can offer you at the moment is to stay humble, courageous, and determined. Do not lie or minimize about anything, and do not get defensive or "blame shift". These are typical responses most WS's seem to find themselves doing, and they often do more damage to an already sensitive situation.
Many BS's find that having a written timeline of events helps them to put the pieces together. Make sure you have one.
Apologizing is a good thing, however, most people respond better to empathy than to an apology. In other words, "I'm sorry I hurt you, I feel really bad about that" is really a statement about yourself, not them. It states how YOU feel. What they want to know is if you understand how THEY feel. So, "I'm sorry I hurt you, it was thoughtless and cruel of me to treat you like that, and I hurt you as a result. You deserved better." is a better approach. I'm not telling you to "just say the words" however, the goal is to change the narrative in your own head, so that thinking of others is a natural thing to do.
You said that your husband is still around, although undecided. That's a gift. That is time to work on yourself and to do whatever you can to prove him and the kids that you are willing and able to do whatever it takes, and that you really mean that. Also, you have to be prepared to let them go. Sometimes, infidelity is a deal-breaker, and no amount of apologies will help. It's important to know and accept that truth.
Keep coming back here. There are lots of good people here who can help. Don't expect to always get a hug a kind response however. Often, we need a little "hard truth" around here and that is often hard to hear and even harder to accept. But if you keep an open mind, and are willing to look at yourself in the worst possible light, then you will likely find that those harsh criticisms often lead to revelations.