Use the date in which you established your domicile.  If you moved out to California but didn't maintain a place there until your wife moved out also, then use that date.  However, if you acquired a residence sooner, then use that date.  In situations like this, the answer to the question really depends on facts and circumstances, which, in this case depends on when you "settled" in to California.  It seems that you have children or at least other dependents, so it does sound that the correct date is closer to when you moved as compared to when she moved after tying up the loose ends from her job.

The dates, however, do not have a significant effect on your tax return:  your Kansas income is taxed to Kansas, and your California income is taxed to California.