Biologically, it is not certain that you will inherit some DNA from each grandparent, great grandparent, etc. At each generation, the DNA is a random mix of the DNA from each parent, each of which are in turn a random mix of their parents. Given enough generations, randomly some DNA from early ancestors may be lost. In fact, there's a 1 in 64 chance that you've inherited nothing at all (or twice as much) from a particular great great grandparent (rather than the 1/16th you'd expect), and a 1 in 8 chance of losing DNA from the generation before that.
Apart from that problem, there is also no way to know the correct ancestry proportions of each ancestor. They may be described as German, but in fact have a Russian parent (so reducing their German proportion by 50%), or grandparent (reducing it by 25%). So you'd need to go back far enough so that the chances of their ancestry being different become insignificant - say 7 generations to bring it under 1%. At that point, the chances of some ancestry being lost or amplified (as described above) becomes quite likely.
This means that in practical terms there is no accurate way to tell what proportions you inherit from each distant ancestor, except maybe through genetic testing of yourself and very many distant cousins.
You're also mixing up ethnic origins and nationality (and both are very ill-defined terms, evolving over time). It's perfectly possible to be Jewish (or Arab, or Romany, or Scots) and have nationalities from one or more countries. For example, maybe that Jewish grandmother was somebody who was born in Germany, grew up in the same place that was now in Poland and later became American.
So in summary I don't think it's very practical to try to calculate an "ethnic mixture", as the source data is incomplete (not back far enough) and subjective (depends on cultural assumptions and biases) and the biology means the calculation can't be accurate.