How can i express how sorry i am to my wife who i have hurt in addiction?
Stop telling her and show her. She has endured ten years of apologies and excuses, she knows your word means nothing. If you feel like you are making better progress this time around, then commit to it and follow through. It will take years of sobriety and consistent good behavior before you can expect to truly earn her trust. There are no words that will magically make this right and no grand gesture that will absolve you. Real change takes time and you need to do the work. In the meantime, accept that your marriage is over and let her move on. Your priority isn't getting her back, it's maintaining an amicable relationship for the sake of your children. That can't happen if you fall back in to your old habits of guilt tripping and manipulating her in to giving you another chance. If you truly care about her and want her to be happy, then you will stay out of her way and let her rebuild the life she wants for herself on her own terms.
If you are not in rehab, get in there and stick with it. And what Kei said. You have a lot of work to do, all by yourself. She obviously loves you, but drugs and alcohol damage your brain. Understand that, and if she is being hurt by living with you, find someplace else to live, and arrange for court custody and child support so you can coparent in peace.
Empathize with her. Saying you’re sorry doesn’t demonstrate that you understand her pain. And if she believes you don’t understand it she’s alone with it. If you haven’t lost her already, another “slip” likely would do it. That said, you have kids to parent. Whatever it is you’re beginning to “see,” don’t abandon your critical faculties in the process. You’d be letting your kids, your wife, and yourself down if you do.
"How can i express how sorry i am to my wife who i have hurt in addiction?" You acknowledge her hurt, and that you are resposible for it -NO excuses. You keep working on getting control of that addiction. You avoid ANY situations where your addiction might be fed - again no excuses. Like you don't go to the Pub with the boys if you know you won't stick to soft drink. To choose to go where your addiction can be fed is to choose the addiction over her and family - KNOW that, because she does. You SHOW her the effort it takes to grow past it, dealing with the problems that got you heading towards addiction in the first place - and there are always background problems to addiction. If you are not already in some sort of addiction program, you join one, and you keep attending. You seek help from medical professionals, therapists. You realise you can't kick this on your own. You accept that she does have the right to say at which point she has had enough - and that YOU own that by your choices. You give her space now, time to show your commitment to getting sober and free of the addiction. You don't use any break as an excuse to fall backwards. You show her that, no matter what happens, you are committed to kicking this demon out of your life - and hers. Its not an easy road, but with time and commitment you can see the end of it. Good luck.
You deserve to find a girl who actually loves you. This current girl does not, as that would save you from addiction.