I thought they will go into defensive mode and that is when they will start loosing wickets as players are not that great in solid defense and some do not have the patience to stay in the crease for a longer period of time without striking the ball and the good news for New Zealand is that Trent Boult will be playing in the next match and then it is going to be a different ball game as England will be struggling if they are not careful.
Boult will boost the Kiwi bowling lineup, but the question here is whom he will replace. Wagner, Southee and Jamieson performed exceptionally well during the first test. Santner is the lone spin bowling option and there is a chance that he will be dropped to make way for Boult. But IMO a better option would be to replace de Grandhomme with Boult. The former is not in great form anyway. But the selectors may be wary, of replacing an all rounder with a bowler (who is also a tail end batsman).