
You say you've had her for 5 weeks, but how old is she at the moment? I do not contribute to keeping a young puppy in a small crate - until the puppy knows you will be letting her out of the crate regularly (during the night) she will just empty, and in a small area, be forced to lie in it!!
How long it will be to be clean all night will depend on the individual puppy. My Whippet came to us at 9 weeks and to be honest, could have held right through although I didn't force her to do this. I'd feed the last meal by around 5 pm and she went into her crate by around 10.30 pm, having been taken outside to empty immediately before going into her crate. I got up by around 3 am, actually anticipating her 'need' so she didn't think she yelled and I came!! We were up to feed by 7 am and during that time, she was clean. Frankie came to us at 4 months and at first, as he'd come from a kennel situation (breeder) I gave him the benefit of the doubt and took him out once overnight too. Very quickly it was obvious that being that bit older, he was physically able to hold.
To be honest, unless your puppy (again age is relevant) isn't sleeping through, once awake, the first thing will be to empty. I'd have her in with you in her crate (shut in) to make getting up and out 'easier'. It should mean you are not greeted with a mess every morning.
Once into a routine, she should learn that you'll be attending to her toiletting needs, and so start to hold (within reason). Young puppies should be given the opportunity to empty roughly every 2 hours during the daytime, unless still taking a nap, and as they are not eating/drinking overnight, only need one outing - assuming the stools are normal!
ps casting my eye back on your answers - what works for one dog, may not for another, especially re crating which is why you see conflicting info. There's NO POINT shutting a puppy in a crate if it's going to become hysterical being in there - and so more likely to mess!!