Let's start with that date, "The 34 February (written upside down) th, yrae (that's right, not year but yrae) 349." That date is written in a way that doesn't leave much doubt about Poprishchin's state of mind at this point.
But let's continue to read. He starts out the last entry by saying something not so batty. Poor Poprishchin is suffering, he knows he is suffering and just wants to be saved. This part almost makes us pity him, right? Does he finally realize he's in an asylum—some insight, maybe? But when he starts to fantasize about being saved, something happens to the writing: "Take the reins, my driver, ring out, my bells, soar aloft, steeds, and carry me out of this world!" (20.1). Just as we were about to start feeling sorry for him, his writing turns… not quite crazy, but really, really fantastic and exaggerated. He even has a vision of his mother in his childhood home.
And then there's that last sentence that takes a sudden and bizarre turn: "And do you know the Dey of Algiers has a bump just under his nose?" Wait, what does that have to do with anything now? What on earth is he talking about? Well, that's exactly it. We have no idea what he is talking about. With that last crazy sentence, Gogol abruptly rescues us from becoming a puddle of tears about Poprishchin's terribly sad fate. And it seals the deal on the question of whether our hero is totally, irredeemably mad.
Gogol could have made this last entry completely crazy, from beginning to end, but he doesn't. Instead, he makes us repeat a mini-version of the whole story, from normalcy to foolishness to pity to madness. Why do you think that might be? Does he want us to laugh or cry at Poprishchin's predicament? Do we have to choose?