Sorry to hear about the problems you're having. It's a common thing I see all the time.
Your partner's ex is happy with the situation and things won't change unless you take action. If you have been unable to come to an agreement court would likely be your next step.
As they are still married they are still linked financially and dissolving this link is something the court can do.
When it comes to Anciliary Relief (under the Matrimonial Causes Act) it doesn't matter who has put in or taken out at any point throughout the marriage. There are many implications here - one relevant one being that the fact he is paying half the mortgage won't be that relevant.
Furthermore, the fact that there are children means that when the `pot' is divided up whoever is the primary carer will receive the lion's share of any equity left over. At any financial order all assets and liabilities - equity, investments, money in accounts, loans, pensions, everything - will be added up and things will be divided. It's common for a parent with the children to get at least 60% of any assets.
Your partner is NOT legally obliged to keep paying the mortgage but as you say that may lead to repossession. That said, mortgage companies are extremely reluctant to do so unless there are no alternatives.
You mention her affair. It's most likely irrelevant. Any court involvement in this issue will be under the Matrimonial Causes Act and not the Children Act. Under the Children Act wishes and feelings are a factor. Under the Matrimonial Causes Act it's more a case of `cold hard cash' and behaviour of the parties involved are ignored unless they are extreme (and an affair is seldom likely to be considered so).
What ultimately happens however is down to whatever the court orders or what you can negotiate.
As your partner cannot afford a solicitor he should consider a McKenzie Friend. They're not solicitors or barristers and don't do exactly the same thing. They don't have automatic rights of audience, are not able to litigate or act as your agent. But good ones are as good as any legal professional and often much cheaper.
If you DO use a solicitor make sure they have experience of contract law, etc. Although this topic is under the ambit of Family Law we're talking about property and contract issues and I've come across more than a few hearings where orders have been made that aren't enforceable under property and contract law. Family Law solicitors are experts in Family Law!
You should also consider joining Families Need Fathers (www.fnf.org.uk/). It runs a helpline, forum, support meetings across the country (many with solicitors offering free advice) and it's not just for fathers - mothers, fathers, stepparents, aunts, uncles - anyone who is involved with family separation.