George don't make it an E mail. Send her a letter but don't finish it yet....
Your wife possible feels that the only way she can hold her head up in the playground and with her friends and family is by ending her marriage and taking back some control. In fact it is precisely what I advise people to do if their H is dithering about his feelings post-affair.
A huge amount of hurt pride, as well as hurt and shock, is what is galvanising your wife at the moment. She possibly feels that you are only trying to get back with her because of the DCs, the finances and the status of being married with DCs. She cannot believe at the moment that you love her for her as AF says.
This is going to take a lot of time and patience on your part, so don't give up.
One of the things you might want to offer is a completely fresh start somewhere else, if that's what she wants. You actually need to offer anything - no change would be too extreme if it means that you can stay together.
As AF says, you need to be sure that you would be staying for her and no other reason. If you're sure now that you love her and don't want to lose her, because of her then that's what you must say, over and over again.
You need to speak to her family and friends too and explain yourself, but not as a way of persuading your wife and getting at her through them - but as a way of apologising properly to them for the enormous hurt caused, acknowledging that affairs hurt far more people than just the immediate family.
I think at the moment, your wife's feelings include wounded pride, shock, anger, disgust and shame. The latter might seem illogical, but because ill-informed people make judgements about why affairs happen, many people actually feel ashamed when their spouse has an affair. The very fact that she is wondering why she wasn't enough for you means that she has absorbed this message.
Like so many people, she has probably been brought up with a set of beliefs about infidelity - that it only happens when someone wasn't getting enough from their marriage, that they are always about sex, that a spouse can keep their partner from straying, that a man won't have an affair if he is "happy at home" - the list goes on and all this will be churning around in her head.
Every pitying look she is getting in the playground, she will be interpreting as judging her and not you. Even members of her own family might be saying things like this - and this is all going to make her feel worse about herself - and therefore you. Worse still, OW and her cronies in the playground might well be casting aspersions about her and her sexual competence. It is all horrible and she will want to do something - anything - that will help her feel back in control and a strong, assertive woman who refuses to be treated this way.
I hope she has got a wiser soul somewhere in her life who will urge her to separate all these feelings and come to a rational decision, based on the facts and not the myths. Someone who will manage to convince her that everything she may have believed about infidelity is nonsense and that there was literally nothing she could have done to prevent your infidelity. That this isn't a reflection on her at all.
That person might gently persuade her that all the understandable feelings of hurt pride and anger often get in the way of a sound decision-making process. That it is better to reflect and work out whether if she tries to forgive, you will do this again (she will be hearing lots of stories about once a cheater, always a cheater), whether she genuinely loves you, whether you want her and not the package that comes with her - and somewhere in all this, what's best for the DCs, acknowledging that she shouldn't be staying with you for their sake alone, just as we advised you upthread.
A good counsellor will be more objective perhaps, especially someone who understands infidelity and signs up to the message of a wonderful book called Not Just Friends, by Dr. Shirley Glass - could you suggest that your wife gets some solo counselling with someone like that?
Once the shock has worn off, most betrayed spouses realise that they have invested a lot in their marriages and that they owe it to themselves (although they will say it is only for the DCs) to try to forgive and get past this. But this has got to come from her and you should not be suggesting that she would be in any way at fault if she calls time on the marriage - accept that you did that yourself, when you had your affair.
You need to create the conditions for her to work again on your marriage, while allowing her to get back her dignity and self-respect. This is where the avowed love for her comes in - and doing everything in your power to make her life easier at the moment; showing your love in actions. It is also where the offered house move comes in ("I would live anywhere as long as it's with you. I don't care about my job, what sort of house we live in, as long as it's with you") and by falling on your sword with her allies.
Have you also been in the playground yourself since discovery - and put people straight? Have you put OW straight, too? Told her that you deeply regret the relationship and that you love your wife deeply - that this wasn't about your wife at all? Told the OW that you will never again befriend someone who wishes your marriage harm?
What betrayed partners look for is a set of actions that back up the words. Strong, decisive actions too - as opposed to lots of hand-wringing and hair shirts, some of which smack of self-pity and self-absorption.
This is a long game George. But fight for your wife. Get her respect back for you and challenge yourself to do things she never thought you were capable of.
Oh that's all we need, a poster who says that infidelity is always a sign that something's wrong with the marriage...I despair. 
