A mortgage is a charge on a property. If you fail to pay the mortgage the property is repossessed. That is why you cannot have a mortgage on a property you have swapped with somebody else.
A mortgage cannot be transferred to another property.
With negative equity it is going to be very difficult to sell the property and get another mortgage.
For you to buy her out would involve terminating the existing mortgage and getting a new mortgage in joint names. This is still going to be difficult - no lender will be happy about giving a 110% mortgage in the current climate, and you will still have to find legal fees, the early redemption penalty and so on.
Talk to the mortgage company.