You sound like a wonderful friend, and very thoughtful too. Your friend must be very tired, so I'm sure anything you can do to help will be appreciated.
Can I ask - how was she diagnosed with PND? What are her symptoms that she has that led to this diagnosis? And what treatment from the health system is she receiving?
I only ask these questions because some health professionals can be a bit keen to diagnose PND in anyone showing any signs of difficulty after having a baby.
Other things I would encourage her to do would be to reassure her that you are there to listen to her if/ when she needs to talk. Don't just concentrate on 'cheering her up' (although distraction and activity are very important if she is suffering from depression) but try to get her to talk a little about her feelings, and if you can, just listen, nothing else. Try not to offer endless solutions, or if you do, try to do this as collaboratively as possible so she doesn't feel she is being 'rescued' but has a sense of accomplishment for herself.
While decent sleep is an impossibility with a little baby, other things such as keeping active, getting fresh air, and eating a healthy diet are all very important. Try to encourage her not to resort to drinking a lot of caffeine, as in the long run this will aggravate her sleep patterns, and leave her feeling anxious (not a great thing if already depressed!).
She should be ignoring comments like those from her mother, they are not helpful. Perhaps you can concoct some amusing retorts together over a (decaf) coffee when you're out shopping next week?!
With the right support from decent friends like you, I'm sure she'll pull through this. I hope things improve for her soon.