Is the mortgage in both names? It is strange that the mortgage lender didn't pick up on her debt.
Has it accrued after the mortgage was taken out or was the info on the application form less than honest?
In order to put the house in his sole name he would need the lender's consent
He could try and work out the appreciation in the 4 years she has lived there and try and work out a fair amount bearing in mind what she has contributed or not, how much of her debt he has paid off etc... and offer that they sort it privately and she goes away with something towards her debts.
The marketplace is not a great place at the moment for good deals on mortgages.
He could ask the mortgage lender (to see if he could keep a decent rate) what the procedure would be to put the mortgage in his sole name, fees, admin, whether they would insist on solicitors being used which would be likely to involve her being advised to take independant legal advice.