I checked, the speaker does not pronouce the final /e/ in więzienie - also I guess the singular verb should have given me a clue.
The pronunciation here is correct. The "e" is pronounced.
I do hear the "e" in "więzienie".
Unless you meant any living speaker?
We have jails, the slammer, the hoosegow, the crowbar hotel and finally tank. Let's see some non native figure that one out. This is what makes it difficult if you aren't a native familiar with the lingo or language as civilized refer it as. Enjoy.
An alternative spelling for "jail" is "gaol".
Why "w" and not "na"?
Both those words have their own collocations, "w" is (I think) a lot more common, 'the default', so to speak. I can't imagine anyone saying "na stolicy".