The facts state that the questioner has already gotten a default judgment against the dog owner. In some states, a landlord may be liable for damages caused by a tenant's dog on or near the rented premises. Generally, the landlord must know or have reason to know that the dog is dangerous. Since it appears that the dog owner may be unable to pay the judgment, pursuing the landlord may be the only option. I suggest contacting an attorney licensed in your jurisdiction to explore whether there is such an option and also whether there is any basis for recovering attorney fees so that it is economically feasible to pursue it.
This answer is made available by an attorney licensed to practice in the state of Oregon. The communication is intended for educational purposes only and is not legal advice. Actual legal advice can only be provided by an attorney licensed in your jurisdiction, with experience in the area of law in which your concern lies. By using or participating in this site, you understand that there is no attorney/client relationship or privilege between you and the attorney responding.
The landlord is in no way responsible for the bill unless you can prove that s/he knew the dog was dangerous and even then, as you have found, winning a judgment and collecting that judgment are two entirely different things. You may want to turn collection over to an attorney that handles those kinds of cases, but even that is no guarantee of payment.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.