Where are your sugars to start with? I'm partly guessing here and don't know this for fact, but 2 things I think might also impact this are increased sugars and increased BMI reducing insulin sensitivity?
Also, if you are taking insulin for just a black coffee, then it sounds like you are taking a correction dose? This suggests you are either running high over night, experiencing Dawn phenomenon or even going hypo and experiencing the symogi effect. If it is any of these, then the insulin could appear to be not working, but is in fact simply countering the raising BG?