I am told that the two languages are similar enough that she would be able to understand much of what is said in Polish.
Totally different. Hungarian/Magyar is a Uralic, Finno-Ugric language, more related to Finnish than to Polish (a Slavic, Indo-European language).
Perhaps if you talk about paprika (Hungarian word) or Gulasz (Goulash... another Hungarian word) you might have some luck. Otherwise, no real relation other than loan words, as far as I'm aware.
As for your Polish Jew understanding Hungarian, it's doubtful. Just consider "Good Soldier Schweik", who is Czech (a language much closer to Polish), who complains about the Magyars and how he can't understand them.
Your Jew might speak Yiddish ( a language similar to German and widely spoken throughout European Jewish communities).
As for your neighbour knowing several languages, it's possible in that way, as a former student of mine in Macedonia lived on the edge of the Gypsy quarter and played with Gypsy and Turkish kids, so she knew a little of each language (Romany, Turkish) and her own, Macedonian, even though the languages are totally different. if your girl already has some familiarity with Hungarian she might find it easier to teach herself. Not impossible.
There is always the possibility, as Poland, Hungary and Czech(oslovakia) were part of the Austro-Hungarian empire, that many different ethnicities could converse in German. Not sure how easy a Yiddish speaker and German speaker can converse, but I can sometimes read Yiddish with my basic German knowledge. German, of course, was also a language of education for many.