when you add ne- to spát, it loses its accent and becomes nespat. Spat on its own is not a word.
"I try not to sleep NOW" sounds very strange in English and I don't think you'd ever hear it.
But there's nothing grammatically wrong with "I try not to sleep" on its own -- or with other information as part of the sentence. For example, "I try not to sleep... when I know I need to study all night to get decent grades on my exams."
Not that this is authoritative in any way, but I don't think I've ever heard that word order. I can imagine "I am NOW trying not to sleep," but "I am trying NOW not to sleep" just sounds strange to my (AmE) ear. FWIW.
"Since I had a 6 AM meeting, I went to bed extra early and tried to sleep. I tossed and turned. Now it is 3 AM and though I wish I could sleep till 4, going to sleep at this point would just make me groggy. I am trying now not to sleep."