In the on change handler of whatever you're hooking up the DatePicker with, do this:

[datePicker setNeedsLayout];
[datePicker layoutIfNeeded];
CGFloat width = [datePicker systemLayoutSizeFittingSize:UILayoutFittingCompressedSize].width;

The first two lines are needed in order for it to be the latest size and not from before the change.

Credits: developer.apple.com/forums/thread/790220