Pumble you're not a shit mother, you're 'just' completely knackered.
I had a male health visitor - who was a former cpn after dd was born. On his first visit, he told me I was at a high risk from pnd and almost talked me into it.
I was given anti-depressants without any real proper diagnosis and nearly 6 months later when I saw a female gp (who'd had 4 children) as I wanted to stop them as all the appts were causing a problem with work, she said that as dd was 4 months old at the time of the diagnosis, I probably just still been quite tired after the birth as I was breast-feeding and had an 80 mile round trip.
I'm not against a proper diagnosis of pnd and treatment, not at all, it's just in my own case, it was probably the beginnings of a seriously underactive thyroid problem, that 15 years later still isn't getting the right treatment for me to feel well.
I know it's a hell of a lot easier said than done, but at 5 weeks, gripe water can become your friend instead of infacol if your dd has evening colic (and you get loyalty points from your chemist to spend on yourself), and eating properly and sleeping enough, in the afternoon when dd1 has a sleep if you can, will work wonders if you are suffering from a combination of tiredness, hormones, with Christmas slapped in the middle of it all.
Be kind to yourself 