I had depression when I was younger and I think I might be able to give you some advice from my own experience.
The part about wanting to die but being afraid of physical pain is something I've been through. I think when people really want to kill themselves they don't even care about any physical pain at all, so this is encouraging and suggests she's not at the point where she really wants to die, yet.
However, you need to realize that this IS a cry for help and you can't ignore it. She genuinely wishes she wasn't alive, she's just afraid to take the step.
She's way over her head and doesn't know what to do. Depression makes you feel like life is too hard for you, like everything you have to do requires too much energy and motivation. As a metaphor, imagine that every responsibility you have in life is a pebble you have to carry, but depression makes it feel like a boulder. It seems insurmountable and you don't know how others find the energy to do it. The key is to rest, until the mountain of boulders seems like a pile of pebbles again.
You should remind her that depression is an illness, it's not her fault, it will get better, that you can manage without her until she gets better, but that you love her and you and your child will need her once she's better.
To help her recover, try taking as much off her plate as you can. Take care of your child, of cleaning the house, etc. The more she has to do, the more crushed she will feel. She needs to rest in order to recover, at least that's what worked for me (I took a year off from school personally). Enlist the help of friends and family if you can, or consider hiring help if you can afford it (part time nanny, weekly housekeeping...). Try finding resources in your area, such as organizations that offer support to people dealing with mental illness, they might be able to help somehow.
It's going to be very hard and frustrating for you too, but try to remain patient and understanding with her, she needs to know that you are always there for her and don't blame her. The perception of having social support (i.e. that people are there to help you) is very important in recovery, so do your best to make her feel like she has all the support she needs. Stay optimistic too when you talk to her, tell her how things will change for the better.
Also, make sure your wife has a whole day off each week for a hobby! I can't stress this enough, it was a huge help to me personally. It was the one day I didn't have to worry about having to do anything, I could rest and enjoy myself as much as I wanted, and it reminded me that life could get better and I just had to hang in there. It didn't cure my depression, but it gave me hope and made me determined to fight it. A daily one or two hour break would also be good for your wife if possible, but this has to be rest time that she knows NOTHING will interrupt no matter what.
Medication can help, but it tends to have secondary effects that can be unpleasant. It can also be unpleasant if you forget to take your pills even one day as it disrupts the cycle. For these reasons, many people hate medication or stop taking it. My opinion is that it can be used temporarily as a crutch, but for most people lifestyle changes are much better in the long term. It's also important to use the right medication, and I'm surprised her doctor won't try another one or modifying the dosage. If the doctor is a generalist, your wife should see a psychiatrist.
I would also advise seeing a psychologist for therapy. Note that a psychologist isn't the same as a psychiatrist: psychiatrists go through med school (they have an M.D.) then specialize in psychiatry, and as such they are able to prescribe drugs but their knowledge of therapy is often lacking. Psychologists study psychology from the Bachelor's to the doctorate, they cannot prescribe drugs but they are much better trained at administering therapy and making patients feel comfortable with them.
Your insurance if you have any may cover one and not the other (usually psychiatrists are covered while psychologists aren't), but many psychologists work with psychiatrists in the same cabinet, which can be used as a legal loophole to have therapy covered (the psychologist administers therapy under the supervision of the psychiatrist, and that's good enough for insurances).
It's also important to find a therapist that your wife will be comfortable with, so she might have to meet a few before settling with one.
With that said, therapy is recommended but it isn't absolutely necessary if you really can't afford it. From personal experience, lifestyle changes (rest) are the most important thing for recovery, and a therapist will mostly try to help her accomplish that. Don't bankrupt yourselves over this and spend your money smartly - you could hire a housekeeper once a week for less than the cost of therapy and it could be just as helpful.
Finally, depression can take up to several years to recover from, but it also depends highly on the support people receive. In my case it lasted for 3 years total, but I spent the first two years without any help and with people forcing me to deal with various life responsibilities. I recovered fully once I took a year-long break from school, and I had already noticed improvements after only 6 months.
The difficulty is to balance resting and the realities of life. You probably won't be able to take care of your own responsibilities plus your wife's. The more of a break she can get the better, but realistically she'll probably have to help you with taking care of your child and some household chores. Just do your best to give her as much of a break as you can.
Don't turn down any help from anyone, if someone offers to help then take them up on it, even if you think they're just trying to be polite and they don't really mean it. Trust me, it's a mistake to refuse help out of fear of inconveniencing people. Ask for something small, like shopping for groceries for you one day so your wife doesn't have to do it. Most people enjoy making a difference in someone's life, and anyway your wife's health right now is far more important than inconveniencing a friend or a coworker for a few hours one afternoon. And really, if they offer help and don't mean it, well they should know better and that's on them.
Also, make some time for yourself to rest during the week because it will take a toll on you if you exhaust yourself taking care of your wife and you never catch a break.
It's going to be hard for you, not just your wife, but remember that if you can pull it off then it's only temporary. It WILL get better for both of you. Talk about all of this with your wife and start planning your life around her recovery.