Just a few quick things:
I sincerely apologize to her daily
Just be aware that words are meaningless to your spouse right now. You also promised to love her and be faithful until death do you part, but those words didn't mean much either, so apologies mean even less. What matters are actions. What you do means much more than what you say. During the affair, your put your own needs ahead of hers, to her detriment. Now she needs to see you sacrificing for her. She needs to know, through actions, that she matters to you. Because your prior actions sent the clear message she does not. (I'm not saying to not apologize, but see it as sprinkles on the ice cream cone, not the cone itself)
But I also need help to make her daily life bearable
As others have said, you can't. Infidelity is trauma, straight up, no less so than a death or a violent accident. If you could afford to hire every therapist in the world, it wouldn't help, because she has to heal at her pace, and in her own way. The very best thing you can do to help her, is to not make things worse for her. No TT, don't get defensive, don't "overtry" to make things better. Be aware that simply wanting her to not hurt... which I'm sure sounds and probably feels very empathic right now... is really more about you wanting HER to feel better so that YOU can feel better. Do you understand what I'm saying there? She might need to feel bad for a while, to go through this pain so she can deal with it, learn from it, understand it, grow from it. So don't put pressure on her to feel better, that might feel like more about what YOU want to her, and that will only lead to more resentment from her. Instead, work on yourself. Go to IC. Go to support groups. Figure out what it is about YOU that ended up with you having an affair. If you can fix the part of you that allowed the affair to happen in the first place, that allowed you to betray her, lie to her, to ignore her well being... fix that, and you will at least be a safer person for her to be around. Right now, she avoids you because you are massively unsafe to her.
is it okay if I kept some distance for now? Because while i'm in the house, all she does is stay inside the room and I want her to be able to move freely while she's home.
Sounds like a good way to put her first, and sacrifice your own comfort for hers. Why is she the locked up prisoner and you are the free one? Why not stay out of her way and give her run of the house? This is good thinking on your part, keep it up. She might want you to stay home because frankly, at least she's knows where you are what you are doing that way. But give her that choice. Maybe even getting out of the house for a week or two will give her some space and time. This is a lot to process.
Some of the things my wife asked me for (which I wish I had been smart enough and not selfish enough to offer on my own in the first place) were:
* Seperating the bank accounts
* Signing a post-nup giving her the lion's share of the assets if we divorced (we lived in a 50/50 state)
* Removing me from her 401K and insurance payouts
Basically, she had already been screwed over enough already by me, and wanted to know she was "safe" from being taken advantage of in other ways. This removed a lot of doubt from her mind in regard to her own safety, at least financially. These are things you can also discuss with her. Our spouses need to see remorse moreso than regret. The need sacrifice more than shame. They need to see that we "get" what we just did to them one some basic level. And they need to see us turning our worlds upside down in order to "make things right" in whatever way we can, in the same way we turned their worlds upside down and made ruined everything.
Last thing - infidelity is often a deal breaker. You could become the world's most perfect person moving forward, it might not make a bit of difference. Glom on to that concept right now, and make sure to tell yourself that it is 100% okay if she leaves. Not for you maybe, but for her. It will help both of you if you can accept that now, and doing so will help you to remain less defensive and less manipulating (even if you aren't aware of it). Most WS's want our spouses back so badly that we often forget that we're simply imposing our own wills on the relationship again. Everything is changed now. It just is.