No matter how much speed Ivan put on, the distance between him and his targets did not decrease in the slightest. Before the poet could gather his wits, he had run out of quiet Spiridonovka and ended up on Nikitsky Gate...
I am sorry to tell you that, but you are way off, and what you give as examples has nothing to do with what's been said.как here is part of the complex conjunction(?) не <...>, как, often used in the expression не успел <verb>, как. They both mean "he hardly did <...>", "before he could". Examples:
Не успел я сделать и двух шагов, как снова услышал его голос - Before I even made a few steps I heard his voice again.
Не сказала она и двух слов, как в зале раздался хохот - She hardly said anything when the audience started laughing.
Note that you can't put a bare verb without успеть, so не опомнился поэт, как is wrong. You have to use a verbal phrase or add at least a particle, for example: поэт ещё даже не опомнился, как.