A dog is expensive, first of all! You have the price of a dog, then the vet, the shots, the food, the accessories, the toys, the licensing, the leash, the crate. And it sounds like you will need doggie day care, either at a center or having someone to come into the house. If you go on vacation, you need to arrange for dog care, which is expensive.
It is cruel and unrealistic to leave a dog, any dog, for those long hours. They are social animals, pack animals, and they cannot be left alone unless and until they are extremely mature and are very used to your family. A puppy cannot manage, and an adopted older dog already has been abandoned or re-homed once. You cannot leave them alone like that.
You should get a pet because you want to give a pet a home and a life and a family, not because you want the dog to have a job (namely cheering up a child). The dog is work. They have to go out multiple times a day (especially puppies - you will be up at 3 AM as you were with an infant), in bad weather and good. You go out in the rain, the heat, the snow. You go out at 5 AM, noon, 4 PM, 8 PM, and before bed. Your daughter may well resent the dog as much as love it, because you will both be stopping what you are doing (dinner, homework, favorite TV show, whatever) to take the dog out, clean up pee and poop, and doing all the training. An untrained dog is an unhappy dog. And an unsocialized dog who is left alone all day can become destructive, chewing on your daughter's favorite shoe or eating her pony tail holders. If the dog gets out, you stop everything you are doing to rescue it.
I say all of this as a dog lover and a dog owner. They are loyal and wonderful members of the family, but you have to know what you are doing, you have to dedicate the time, and you have to put in a lot of time to train the dog so it knows what you expect of it. It's not just something sweet and adorable to cuddle.
I am so sorry your daughter is going through so many losses, and it does sound like she is depressed. But a child who wants to get in bed is not likely to want to go outside to stand around and wait for a puppy to learn where it's supposed to pee and poop! I do think you have to address her losses and her emotional state, but I really don't recommend a puppy at all.
If you absolutely insist on a pet, I would go to a good shelter and find an adult cat who is good with people and likes to be cuddled (not all cats do - some are aloof). But cats can be more solitary, they can and should be litter-box trained, and they can be left alone if you have enough equipment to keep them occupied so they are not destructive. If you go away, a neighbor can come in to feed them, provide water, and clean the litter box. But any good shelter will require you to fill out a form indicating that you know how much a pet costs and that this is a lifetime commitment. When we adopted our dog, we did an 8 page application and questionnaire!
The other thing you have to be prepared with for any pet is that, sadly, they die at some point. So be sure you are ready for a loss. You have to be realistic with anything like this!
I wish you and your daughter well!