Usually joint debt liability is outlined as 'jointly and severally liable' - this means the lender can go after either or both of them, so if the ex is defaulting on her share your partner should do his best to keep payments up to date to keep his record clear, but obviously pursue her for payments towards it as well.
In relation to the house, do as bednobs says and arrange your own joint mortgage with your partner, so you can both 'buy it off' the ex, for want of a better word. As she has agreed to do this, it shouldn't be too problematic fro,m her point of view, although you will of course need to be eligible for a current mortgage deal to be able to do this. I suggest you talk it through with your solicitors.