If the mortgage is in joint names, they are jointly and severally liable for the whole amount, not half each. ie. they are liable jointly, but they are also both liable as individuals, for the whole amount.
She should contact the Council tax office as she can get a reduction for living in her house as the only adult - think it's 25% - from the date he left.
She should be able to get tax credits of some sort. And benefits, but I don't know much about that.
If there is an affair, it makes no difference to any settlement. If it goes to court, it's my understanding that they start with a 50/50 split of everything, and then work from there. The primary care of the children is important to this division, as is any decision that she and her h made about her not working to bring up the kids. But it will also be taken into account that he too needs somewhere to live.
Check also whose name is on bills etc. A friend of mine has had no end of trouble after her H left her, and then cut off their phone, electricity, gas, internet etc.etc. All were in his name.
Joint bank accounts - I would get her name off them, so he can't go and spend any money/take the accounts into overdraft - as with the house, if it's in joint names, they will be jointly and severally liable for any overdraft.
Get some advice about maintenance. She will be entitled to child maintenance, and she may get agreement on spouse maintenance too, ie. to support her.
If she's strapped for cash, get her to look for a solicitor who will act using legal aid.
I'm sure there's more, but tell her she needs to get advice, and it doesn't have to be expensive.
Hope she's ok.