I am not sure what to do, I know of multiple imputation, but my understanding is that doing this with non-interval data is problematic (actually I stopped studying this because I was told that on this board years ago). All my predictors are dummy variables, my DV has two levels.

We are doing this to determine which variables are relatively more important, the way we do that is see which are statistically significant (I have found no good way to address relative importance with logistic regression). I am not sure what to do with so many missing cases.

Is it reasonable when you see an unusually high number of cases missing to remove a question, because you think people did not understand it, or had no answer (in honesty I think this is true with the specific question even ignoring all the missing questions - no one asked me about it when it was created)?