I am convinced that the dog owner is responsible for any damage the dog does. Citizens Advise Bureau can tell you for certain. If they refuse to pay for the bills (the dog owner, not the landlord) there is the Small Claims Court for exactly those cases. No need for solicitor or anything like that.
And I agree with TG, it looks like this dog is neglected.
In the first instance you need photographs to prove the case. There must be gnaw marks and the like. Or pictures of the dog in your garden. If the dog can get out and into your garden, the dog is a danger to the general public.
Good luck