Here's the reality you've summarized:
1. She has not invested in the marriage. This is the biggest issue causing her dissatisfaction, rather than anything you've done or not done. And if this doesn't change, she's just another opportunity away from cheating again and again and again.
2. She has everything taken care of for her - the children have a nanny, she has too much time on her hands, and she has a cushy lifestyle in which she has chosen to be dishonest and selfish.
3. She has no remorse for betraying YOU and her family, only remorse for herself. It seems that in her mind, she's the only one she violated.
4. She feels she's in a position to make demands of you, lacks empathy and in fact, is a miserable roommate.
Since these four points exist, regardless of the young age of your children, you have little choice left between either complete self-sacrifice, or sever households from her.
She has had NO LOGICAL CONSEQUENCES for her betrayal of her family or you.
She still lives in her cushy lifestyle, and is just an opportunity away from cheating on you again.
Take away the cushy lifestyle is your number one requirement.
Walking down the path to divorce will give you several things:
1. Logical consequences - if this, then that are the best teachers you can give your children. If you're always there, softening or taking away the blow from her choices, it stands to reason your children will be raised to be dishonest cheaters too because if you'll protect your wayward wife from her own behavior, with this kind of soul-destroying behavior against you, it stands to reason you'll do anything to help your children avoid their consequences. She's spoiled!! Don't do it to them.
2. Leverage. You have the means to gain custody - you have the nanny already in place to take care of them. She can go live the life she wants. Or she can provide you with a plan to invest in the marriage and protect it. Without that investment from her, SHE.MUST.GO!!!
3. Post-nup. If she's willing to invest in the marriage, she must be willing to do it for the long stretch. Demonstrate that she will protect the marriage and intends to do that forever.
What is her plan to make this right with you? My guess is she's not offering anything. She's making demands, which means she has no intention of doing anything but being a spoiled brat.
Is this who you want raising your children?
Talk with a lawyer and get the ball rolling. Let her know she has a limited amount of time to make the necessary changes. That you are willing to be there making changes right along with her so that it's a marriage, not a dictatorship that doesn't care if she's miserable.
But right now, she's just a spoiled brat who had it too good for way too long.
Love is not a feeling. It's an action. It's a result of putting oneself into the relationship - it's an investment of all one is to the care and protection of another.
She doesn't know this. And if you don't recognize her withholding her investment in you and the family, you're asking for a lifetime of pain. Stop propping her up and aiding her in not investing.