You're on the hook for funeral costs. You are responsible for any shortfall on your father's home. That's tough on you, and you have my sympathy.
It needs to be established who is to act as executor. What does the will say? If you are a nominated executor, you can take out probate yourself without involving a solicitor. It's a bit of work, but it's manageable if you are comfortable with forms and jumping through procedural hoops. Having a Grant of Probate could be very helpful in dealing with the mortgage lender, because if they proceed without your co-operation, you might get a worse deal than if you worked with them towards a resolution of the problems.
Which might serve your interests better: to take over the house and continue paying the mortgage, or to sell it soon and take the loss? [Have you a home of your own?] If the house is to be sold, I would certainly follow Brendan's advice and try to make a deal with the bank.
Are there any good bits to alleviate the gloom, things like a deposit account, or a few valuable items that can be sold? Or are there other debts hanging over the estate?