Lack of sleep and iron deficiency will be playing a HUGE part in your current feelings. I'm a useless wreck if I'm sleep deprived, it's horrible and you have all my sympathy.
If I'm desperate for a bath, I put my 3mo in his bouncer and keep it next to the bath - he can still see me, and most times he nods off because it's warm in the bathroom.
I'm not a single mum, and I have the utmost respect for women who are. Kids are relentless. My DP is out of the house working two jobs for 13hrs per day, whilst I'm on mat leave.
The other week, the baby was a nightmare all day, from 3am. Nothing I did for him was right, all he did was shriek, I was already in agony from an abscess on my wisdom tooth, and absolutely shattered.
The baby then whacked me in the jaw with his head, on the side with the abscess. I stuck him in his cot and sat on the toilet sobbing for 10 solid minutes. I was seriously questioning why I'd had a third child, despite the fact he was planned and very much wanted.
My point is, none of the things you're feeling are wrong. You're overwhelmed, and it's all too easy to slip into depression.
If you're in the financial position to put your DD in nursery, go for it. Preserve your sanity, and please PLEASE don't feel guilty about putting her in childcare. The benefits will be immense to both of you.