You would get more responses by putting this on the Pronunciation Forum.
[edited to add: I'm assuming here that the student can actually tell the difference between the two sounds. Some of my previous students (Spanish speaking) could not hear the difference in vowel sounds between /bit/ and /beet/ and the same thing occurs here in Japan (/bit/, /beet/ but also /play/, /pray/). You might want to test first to see if she can tell the difference. Make up a card for short U (hut) and a card for short O (hot) and switch between minimal pair words like hot and hut. The student raises the card of the vowel sound she hears. If she cannot hear the difference then it is a very difficult thing to teach. After being in America for so long she can probably hear them as two different sounds, unless she actually lives most of her life in the Polish language (mostly Polish speaking friends, doesn't actually use English much etc.)]
The short U sound (Hut) (sorry no IPA right now) is pronounced almost the same as the short O sound (hot) except that the mouth is not opened as much for the short U (Hut) sound (in most North American dialects/accents).
Get her to really lengthen the short u sound in hut (like "Huuuuuuuuuuuuuuuuuut") (but of course it will sound like 'hot', "hooooooooooooot") and tell her to slowly close her mouth a little bit while pronouncing the vowel (but not too much) and you should hear it change to Hut (it should sound like 'hooooouuuut' (but with no gap betwen the sounds). Usually I find that it's easier to drop in articulation rather than raise, so if it doesn't seem to be working, then start at a rounded O sound and drop in articulation (boat->but->bot) (as in, slowly lower the jaw [booouuuuaaaaat]). Just make sure to lengthen each vowel a lot, and tell her when to stop. Then it's a matter of her practising until her mouth remembers where to go.
Here's a chart of vowel articulation.
http://www.arts.gla.ac.uk/IPA/vowels.html
The short U (hut) sound is the upside down V at the far right near the bottom. The short O (hot) sound is the lower case 'a' without the top part immediately below the upside down v (I'm imagining an American English accent which is similar to my own English, Canadian from Toronto, or to television or to people I know from the States).
BTW I wasn't sure what you meant by the "th" in "the" and "thin", they're two different sounds (one's voiced --the--, the other not --thin--). I think by double l you mean the L sound in which more than just the very tip of the tongue is pressed to the alveolar ridge (front part of the roof of your mouth). But I'm not sure. I somehow don't think that it would be a major problem for others to understand her spoken English, though.
The "oo" in "look, book, cook, took" (North American English) in the chart is the one that looks like a horseshoe (near the right, near the top). It's harder to explain than the short U. Maybe the easiest way is to say to close her mouth a bit from the long O in "Boat" and push her tongue a little bit farther forward. Sounds a bit easier than it really is, because with vowels the tongue doesn't touch another part of the mouth so it's hard to know where it is at any given time. It's sort of half way between the U sound in "boot" and the "Ow" sound in "Boat", but with the tongue a little bit closer to the front (articulated closer to the front).
For yourself, you may want to go out and get a book on phonetics, it should help (at least it has helped me). And keep in mind that many people never totally adopt a new accent (in this case an American one), and that there's nothing wrong with having 'an accent', especially if there's no problem for anyone understanding the language.