I was once in your shoes, leave her. Every second spent feeling sad, unconfortable, hurt, unhappy, betrayed is your life the only one life you have being wasted. Trust me when I say that I know what it feels like. I felt like my entire life lost its meaning when my best friend and fionce and of 6 years cheated on me. Everything, every plan in my life, every objective, every thought was created around her and I. Your wife may have border line personlaity disorder, mine did. She would have kept it from you if the wife of that man had not found out. These blows damage a good man and right now you probalby have a difficult time even forming thoughts, the thought of such a dractic change, all the uncertainties. But trust me mate, focus on your health and your craft, the fullfilment of becoming the you you dream of is more fulfilling than any relationship, to many of us let relationships define us. When you get where and become who you want to be you wil meet someone there. Or even before, I joined and online dating site eharmony and match with no expectations, created a great profile saying eveything, feelings, objectives, what I can offer what I cant, why I feel the way I do, what Id love, what happened to me etc and now Im with someone how makes me feel as though my X's infedility was a gift. I promise you, move on. Sometimes love is not enough, Loving someone and being in love is two different things. Only until a person who feels the pain of loss after cheating will those impulses of thrilling new attraction and mystery be acompanied by such a deep fear of loss that those feelings instantly turn to replusion.

She had endless opportunities to remove him from her life and her from his and she did not do it. Do not waste anymore of your life on her. She knew she was betraying you. You were facing the same difficulties and were not drowining your issues in indulgencies and sex or lusting after the thousdands of gorgeous woman every where, you upheld your vows. She did not and now she must face the consequences.  Fix you, work hard, work diligently, seek support. There are quite a few support groups in major states.

In time you will relaize that a new life will take shape.

Don't give up first, try to check yourself first. Every problem in marriage is involved two people. If your wife cheated on you, you should think first why did it happen? Maybe she really stressed and feeling insecure? And suddenly this jerk coming and try to get your wife? Many possible reasons dude, but again you are the one who know your wife the most. We're here has no right to judge, but from your story i still can see that there are something that you still can fight for. You have children and you have a wife that brave enough to confess her mistake. At least do appreaciate her efford. 

Dude i will easily say to divorce her, but in your case maybe you still able to work on it before take the final decision. NOW focus on your carrier and your availability for your family. Your working hour don't help at all , it makes you both lack of communication. 

Don't give up, back to your sense and open your laptop start to look for better job and get side job if you need it. But at least commit 1-2 days holiday in a week for your family and try so hard to look for job in that the workinh hour is in appropriate time. 

Try to spare your time for her and ask anythinh that still you hold inside and don't forget to pray in every action that you take. Let you get the wise decision at the end

Okay, you and your wife are very young to have three kids and a big ball of stress. That is enough to make people do stupid escapist stuff. I am old enuogh to be your father, so listen to me: you do not have just a damage wife that broke her marriage vows, you also have three kids that when you fathered them and brought them into this world you made each a Vow. That vow was the act of fathering them. It may not have been verbal but it is more powerful than words. You have to do the best for your kids. That means different things for different situations. You have to figure this out for your situation. It sounds like you need to stay married to your idiot wife. Worse, you need to show her respect and even some love for the kids sake. You may find that your relationship never recovers to what it was before, but this is common for most marriages even without cheating. So, YOU do the right thing! Take care of your kids and their mother. Explain that her misbehavior stops now. She has to tell your family and her family what she did so they can keep an eye on her. The new rule is that she has to clear her schedule with you. Her freedom to shit on her family is over. Then you try to find a better job. You ask for help! When the twins make it to nursery school send them full time and get your wife a job. Hold the shit down. Spend a lot of time with your kids, you never know what the future will hold. When the youngest turn 18, if things with your wife are still shit, flush the turd. PS be careful with your finances in preparation for a possible divorce.  

I cried

Update.

Since my last update, things have gotten worse. At first it all seemed okay, I told her I wanted to focus on building up our friendship before working on our marriage and she kind of agreed (not really) but she totally changed her attitude and was being really nice. We agreed to tell each other EVERYTHING and we we're working on effectively communicating our feelings. She always made comments about how she felt like she was just getting stringed along. We went to marriage counseling, enjoyed some time doing some activites as friends and even went on a date which was fun. But things took a turn.. On the same day we went on our date (about 4 days ago) I agreed to stay at her place so we could all sleep together in the same room (the kids and us two). I woke up after an hour of sleep and had a bad feeling, the same kind of feeling I've had before. So I decide to go with my gut this time and go through her phone. I found a pic of her and one of her guy friends together at midnight on a day I was at work, she never told me they hung out so I go through my messages and find our conversations from the same date (also about a month ago) and it was a night she text me and said she was tired and was going to go to bed early and she never wrote me back or answered my calls afterwards. That's not even the worse part... I went through her email and found that she was still talking to this piece of shit dude she had an affair with... When she swore she cut all ties with him! I'm devastated.. the pain of betrayal was there again and now I'm done. It hurts so much to let her go but I don't deserve this, there's absolutely no trust with this woman anymore. I'm focusing on getting my head straight and becoming a better person all around after this experience, and then I will dedicate my life to my children and be the best dad I can be. Also want to add that I contacted the police department where this douche bag works, told them what happened and how this guy abused his power to get at my wife, and I had an interview with the Lt. of Internal Affairs this morning. He told me this situation makes the Chief sick and that they'll do everything they can to resolve this issue. Hope he gets what he deserves. I still feel really down, but I'm feeling a lot better than the past couple of weeks. Thanks everyone for the advice.

update please