I'm so sorry you're going through this and your dc are struggling so much. My husband of 22 years left in November after I discovered he is having an affair with another officer in the TA (serve your country, destroy a marriage!). Well I threw him out. His parents are divorced and still on speaking terms so he assumed I would react the same way - how wrong he was.
My boys are older - 19 and 16, eldest at uni - same uni as affair partner's son and my son is friends with hers - what a mess. Husband doesn't keep in contact with youngest between visits to help him with revision once a week - youngest has Asperger's and Dyslexia and sitting his "Highers" (Scottish A-levels) this year so needs a fair amount of help. I've explained to both boys that I am going to be upset and indeed they've seen me very upset, but have assured them that I will be ok in the long term, but there's no quick fix.
Like you I have no close family - parents both died over 10 years ago and I have no siblings. The week I found out about affair was the 10th anniversary of my Dad's death - not the best week I've had. I have found that my youngest's mood is affected by how I am. If I wobble, then so does he, so I just need to keep reassuring him that it will all be okay in time. He doesn't cope well with change, but in many ways the house is a nicer place to be with just the two of us in it and we rub along well together. Husband is a control freak - probably has Asperger's too, but also some sort of obsessive personality disorder - not the easiest to live with.
My employer has been amazing and I've only missed a couple of days at work as I find the distraction of being there keeps me going - making sure I see someone every day - even for a short time makes a huge difference. I only work part time so I know I'll need to get more hours eventually, but at the moment tax credits are making things bearable.
My friends have been great and I try to go out for a walk with one friend a few times a week. Obviously it's not so easy when your children are younger, but if your BIL is willing, maybe he could watch your children while you get out for a bit. Try to eat a little - I think I survived on tea and bananas for the first month, but gradually things will get easier. I do have awful days still - had to leave work on Tuesday - but I can see a way forward.
Take all offers of help from friends - they will keep you sane. The advice about being a team is spot on - my boys know I am the anchor and will always be there for them. They see their Dad, but don't seek him out - they wait until he makes contact. I get all the calls from uni about changing flats, all the documents about being guarantor for rent etc (not sure thats a privilege!) - I know eldest trusts me where he wouldn't trust his Dad to do stuff like that.
Look after yourself, visit your GP - even just to touch base and make them aware in case you need help in future - keep children's school in the loop - my son's pupil support teacher has been amazing as son has problems putting things into words but this teacher knows him well and knows just how to talk to him - he's the most amazing teacher I've ever met and has made son's life so much easier over the last few months. Try to keep normal routines and activities - distraction works for the kids too. I also did practical things by email where possible as phoning was difficult - council tax, banking etc.
Take care and know you're not alone.