Did you cheat on her?
Did you sleep with another woman?

Why do you feel like you have to prove yourself to her?
Why do you feel like you have to jump through hoops to get her back?

You don't.

She had the affair.

The fact that you allowed her to move back in, shows you are tolerating her crap behavior just to get her back.

Plain and simple, you need to make it clear to her that you're not here to play these games with her anymore.

"I don't know what I was thinking taking you back after you cheated on me. This isn't what I want. You are not the one taking me back, let's be clear about this, it was me considering taking you back and I can see now that I made the wrong decision to let you back in my life. I've been thinking about this for quite some time already, I can't have someone who acts the way you do back in my life. I'm not attracted to you or any woman who would do this to me, I don't want this and I can't just stay with you because of the kids. This is the wrong thing to do. You don't have the right feelings when it comes to us and I'm starting to see that I don't have the right feelings when it comes to you. This idea & attitude you have about "taking me back" doesn't work for me. I think we should separate again, tell the kids things aren't working out."

Seriously bro, when a woman who cheats on you, has affairs, sleeps with OM, etc. has the attitude that they're taking you back when they were the cheating spouse, nothing good can come of this. You can already see that you're arguing and bickering over everything all over again. She doesn't have the right feelings for you because she can't have those feelings until you man up and really let her go and let her believe she is losing you. She doesn't have that feeling, you said it yourself, she has the attitude that she is taking you back, how much value do you think you have in her eyes when she thinks like this? She feels like she is settling for you when she says she is "taking you back", definitely not something I would consider flattering to hear.

That's your decision.

You can either...
- Take her crap behavior and attitude,
assume the broken spouse role and allow her to take you back so that she can screw around again in the future (not if, but when she does this) and she can then let you go again and you guys can separate and possibly repeat this song and dance all over again.

Or...
- you can let her go, tell her things aren't working out, you don't feel the right way about her and you're not sure if you can feel the right way anymore, you're forcing yourself to settle for someone who doesn't value you or the relationship they have with you and you know that nothing good can come from this. She isn't sorry or apologetic, she doesn't view what she did as wrong because she has a piss poor attitude about you and what she did and she gives you attitude. Big deal, she's being transparent:
"here is my cell phone unlocked, read it as much as you want, I didn't tell you about the pay as you go cell phone I have with no contract that I use to contact my potential other men, that phone is locked, that one you can't look at because it's hidden but let's not talk about that, it doesn't concern you"

It's decision time in your life,
it's time for you to be a man and decide what YOU want,
do you continue enabling her crap behavior or do you set her straight. You don't offer her things like "if you don't smarten up I'm outta here" because that's an empty threat and she'll continue pushing you and pushing past your boundaries and continuing to disrespect you. You tell her that it's over, you need time & space from her, life is short and you've been considering the single life and it's time for you to spread your wings and see what's out there. There's thousands of other women out there just as good or better than your wife, women that would treat you better than she currently does.

But between you and me,
you won't do what's required,
you will complain about her attitude,
the crap behavior, "her taking you back" when she cheated on you, etc. You'll wonder why she can't love you and be attracted to you and continue "trying" with her, doing everything, being Mr.Perfect Husband, helping out with everything at home, working a decent job, being a good provider, being the perfect dad and continuing to accept her crap behavior of you because you "love" her. Let me know how that's working out for you.