I agree with the consensus that gifting a dog is not always a welcomed surprise. You might consider:
1. Can you handle taking in another dog?
2. If so, can you 'capture' the dog, take her to the Vet, house-train her, 'socialize' her, teach her basic obedience, and teach her not to bark?
3. While you are doing this, your brother... will see the dog, with no pressure of responsiblity... friends with benefits

4. Then, you can decide if you want the responsibility of getting her spayed ... and then offer her to him when she is fully healed.
5. He may decide that he is not ready to own a dog... but I think you'll see a minor sadness, while she's being spayed (this is a good sign... different from depression) and a relief and elevation in mood when she returns... and then when she is healed... And all of that may help to kickstart him towards recovery
6. However, I have no other suggestions about what to do if he is 'scared' to take on the responsibility ... except to adopt the dog yourself, allowing visiting rights, maybe encouraging him to slowly train the dog, have overnights with the dog, and so on.
7. I can see how the 'anonimity' and lack of responsibility might be attractive... and it may eventually help to move him forward.