Firstly, there is no "handing back keys". The bank have never owned the house. They own the debt, you own the house.
If you sell the house for less than the mortgage (which you would need their permission to do) you still owe the outstanding amount.
If you hand over the keys to the bank and get them to sell it, you still owe the difference, which would likely be greater than if you sold it yourself.
Basically you have to repay the full amount that you owed plus any interest due. The bank is unlikely to allow you to sell the house for so much less than the loan amount as you cannot discharge the amount. It might be possible to get a loan for the difference but I would seriously doubt it for that much (even if the lending climate was not as cautious as it is now that would be a huge amount of unsecured debt). You would have to have the mortgage fully discharged before you would be able to sign the property over to another.
The only viable option that I can see is perhaps of you can consider selling the other, unmortgaged property and using the proceeds from this to reduce your current mortgage. That may not be feasible or palatable though and I am making an assumption that it is mortgage-free (though you do not imply that and I think my reading would be that it is not an inheritance or a gift but rather it would involve you buying it on favourable terms).
If you walk away, your credit history will be compromised and you will not likely get another mortgage until both the existing loan is repaid and your credit history is repaired (5 years after the debts are fully discharged).