Where does she come from please?

That intonation is perfect for someone who is lazily reading some totally random stuff on a random page on the internet, so let's forget about intonation and rhythm (it is unnatural, but several people read like morons if they don't concentrate).

<<"Stumbling", for instance, sounded more like "stahmbling", which is a U-sound that exists in no variety of American accent and immediately sounds like a foreign mistake.>>

I don't think the vowel in "stumbling" in itself was too weird at all, although I recognize she's probably not consistent in the vowels she chooses, and that one does sound close to others that are supposed to be distinct. I noticed she doesn't pronounce final -ing's as "eeng", so she can't have the Cali vowel shift.

Conclusion: if she's not a native speaker, she really fooled me anyway (I am not a native speaker but I am improving), and if she doesn't sound like a native speaker to the natives, it must be because of some unexpected features in the expected accent, and not because of the presence of non-native features (apart that "speesh"). That "ch --> sh" would be the only thing that could make me think of a non-native feature, but I am not sure. Maybe it's a speech defect or something I don't know. I hear "ak-sint" all the time, so I really couldn't take her pronunciation of "accent" as non-native. This guy seems to say "ak-sint" too (0:20 - ...when they try to do a New York aksint...)
http://www.youtube.com/watch?v=apyL9wBWvIk

I know she's probably a non-native speaker because she's the one who posted it and who is learning English, LOL... but... if you are not a native speaker, I would say your accent is excellent anyway. I guess less than 3% of learners would have an accent as good as that, or better.

For example, one non-consistent thing I noticed is she said something that sounded more like "recog-neshn" at 0:03, but "recog-nishn" at 0:14. That's not consistent, and that's what might label you as non-native, but it's not the vowel quality in itself that is non-native.

Native English Speakers you should calm down,:-) do you know that you hardly lose your heavy accent when you speak a foreign language?

"Stumbling", for instance, sounded more like "stahmbling"

yup, so Australian

drunk [drank]
sun [san]

Realistic - I think you misunderstand; no one's trying to put the speaker's accent down - it's a really great accent for a foreign speaker. The question is what marks it as foreign. If it was indeed the person posting who recorded the piece, I imagine she would want to know. I did this myself nearly a year ago on another forum (posted a sample clip anonymously to see what people picked up on), and all the stuff people pointed out was helpful to me, even if the exclamations of 'that's not native!' crushed some of my cherished illusions.

And Johnny, yes, there are clear non-native features there. They are very subtle, which is why, as a non-native speaker, I gave her the benefit of the doubt in case I was mistaken, but the native speakers here corroborated it. Very good pickup on the second 'recognition' - that's the first word that threw me off. I've never, ever heard a native speaker speak the word like that, even when misspeaking (both the intonation rising mid-word like that in that kind of context and the vowel you pointed out). And the choppy 'has hit a stumbling block', with the broken-up rhythm and the way she handled some of the vowels, sounds quite foreign.

We can't "just forget" about intonation and rhythm. It's a big deal, here and elsewhere. My own voice used to move up and down a lot, and I used to speak a bit choppily, both of which were strong give-aways of my accent. In this particular case, I'd say the intonation is pretty good (except on that second 'recognition'), but the rhythm (both within a single word and within a phrase) gives it away.

Accent is not just phonetics. If it were, learning to speak like a native speaker would be much, much easier, because you would only need to learn how to form a finite number of sounds perfectly. Unfortunately, a lot of the real magic is in what happens when you put those sounds together.

<<Native English Speakers you should calm down,:-) do you know that you hardly lose your heavy accent when you speak a foreign language? >>

We're not upset that she has a foreign accent or unaware that we would have one in another language, too; we're just saying that we can hear it. And people can argue about all of the pronunciations we've pointed out as sounding odd to us all they want; it still doesn't change the fact that she just didn't quite get it right to our ears.

<<"Stumbling", for instance, sounded more like "stahmbling"

yup, so Australian>>

Well, when she tries her Aussie accent out, maybe she can use it there.

>> I noticed she doesn't pronounce final -ing's as "eeng", so she can't have the Cali vowel shift. <<

That actually doesn't have anything to do with the California vowel shift. It's just a regional feature that some Californians have.