Zn Zn++ + 2e- E0=0.76V
-0.76V
Therefore, [H+]=1.89x10-4 M
Wasn't that hard.
Note, that most of the work you did by treating whole system was unnecessary. You could treat Zn electrode just as a reference. As [Zn
2+] = 1M electrode in the standard state and its potential is -0.76V. That means potential of the hydrogen electrode is just -0.22V (vs standard hydrogen electrode), and what you have is just a simple concentration cell.