Tell an agency what you want and they will find you someone. Yes, anyone can call themselves a nanny but only crap agencies take someone with no qualifications or experience. Eden, Greycoat, Imperial - all have reputations to uphold and won't mess around with you.
Say you want a current first aid and conpleted food hygiene. Say you want a level 3 qualification, ideally with additional courses. Say you want x amount of experience and y of that with newborns, or you want someone who has worked with a newborn and 5yo before with a post CS mother with PND.
When you get candidates through just check the CV quickly - you're looking for CACHE DCE, NVQ3, BTEC, NAMCW or NNEB in terms of qualifications. If you need to then post on MN saying 'can anyone experienced look over this CV for me?' and send them an anonymised version c&p in a PM.
But first sit down and think what you need them to do and when. Believe me when I say your AP is no good for what you need and you can find a better solution. Her wanting to stay shouldn't be a factor - you need what you need and if she can't provide it when you thought she could she has to go. She really should be able to keep your DD amused in that after-school slot.
If blondes is offering then grab her with both hands! It might cost more on the short term but you'll need less help in the long term because you'll be able to recover. I'd rather pay 4x more per week for a good extra pair of hands FT over 3 months than a year of a crap AP who also adds to the living expenses of the whole family.
It may not be help every day you need - Tuesday and Thursday full days might give you the time to rest to cope with the other days. Can the Norlander do any more hours assuming she's any good?
You do have to protect your MH but going round the carousel of childcare options isn't helping you do that. It's giving you another thing to worry and get stressed over. Is your psychiatrist managing your anxiety? I think the worry about getting no sleep is almost worse than the lack of sleep for you at the moment.