I am so very sorry this has happened. I can see how much pain you are in.
You say you can't believe "a friend would do this" - what do you mean? Do you mean he was your friend, or do you mean the girlfriend was once your friend?
He moved out 1 year ago, right? So this is not new. Your marriage ended 1 year ago. Whether he should have fought harder for your marriage and family is now a moot point - he didn't do it, and a year later, he hasn't changed his mind.
After a year, he has a girlfriend. Even if he was with her when he moved out (which I'm not saying he was, and you didn't specify), he waited a full year before introducing her to the children. That's commendable even if he was awful to you in other ways.
Your children are not "now" in a broken family - they have been in that situation for a year. You have not "failed completely" - your marriage failed, for whatever reason or reasons that was. But don't take all the blame on yourself. This woman is not moving into your spot. Your spot as their mother remains intact and will never change. She is your ex husband's girlfriend, but that ceased to be your "spot" when he left.
Why do you say she is "trying to take your kids as her own"?? Because she shows up at some events? That is a huge jump and one which is not justified by the facts. Unless she is telling them to call her "Mommy" and saying that you are a terrible person, she's not doing anything wrong.
You must stop focusing on her. She is not the problem. She is the result of the problem, which is a marriage that broke up and one partner (your ex husband) who moved on. The other part of the problem is your inability to accept that your marriage ended a year ago - you still haven't accepted that he's not coming "home" and you are denying the reality that he has another home now. Your marriage was probably over before he left - whether you accept that or not.
Look, people don't walk out on a healthy marriage. If a marriage breaks up, it's because there was a problem. That doesn't mean you caused the problem by yourself - but it's evident that you didn't see it coming or accept it when it happened. You are placing blame on this woman, and not seeing your ex as part of the problem (or even the entire problem, if you feel you had no role in this at all). So you act like you are "supposed to be" with your husband - but clearly you are not. That ship sailed a year ago, maybe longer. She did not move in on a healthy marriage and destroy it - a 3rd party cannot do that. You absolutely must STOP focusing on her, work on your own self-confidence, and deal with your ex in a productive way as a co-parent.
If you don't have a legal separation agreement (including child support and custody), get one. Get counseling to figure out why you are clinging to a fantasy that he will come back, and why you are so insecure that you say 2 contradictory things: 1) that YOU failed and 2) that SHE is the problem.
Your children deserve and need a confident, strong mother, who shows them how to live with backbone and independence. They do not need a mother who, after 1 year, has not taken care of herself enough to get counseling to learn how to deal what may be a very unfair situation. Even if your husband was 100% at fault in the demise of your marriage, you are 100% at fault for not moving forward, for not finding a good grounding and a strong sense of herself. I don't know the genders of your children, but if you have a son, he needs to know that women are strong, deserving of respect, and resilient. He needs to know that a woman is a partner who can enhance his life, and not a victim dependent on him. If you have a daughter, she needs to learn that SHE is responsible for her own happiness and strength, and that she does not NEED a man to make her a whole and complete being.
Your husband may have been a total piece of garbage, I don't know. But even if he was, he waited an entire YEAR to introduce his children to his girlfriend. That is extraordinary and commendable. That shows great respect for the kids, and (frankly) great respect for you.
What you MUST do is hold your head high at school events, trust your position of honor and authority in your children's lives. And I'll go a step further. You will make your children's lives better if you allow for the possibility that another woman can make them feel even better about themselves by showing them love and caring. She could be your greatest ally. You should want your children to be loved by as many people as possible.
I am eminently familiar with step- and blended families. I urge you to take the high road here, from a position of confidence and strength, and be the biggest, proudest mom you can be. All you have to do is to reach out for professional help and support to get there.