This is a tricky question.
As strong evidence that at least some occurrences of "go to sleep" are using the preposition to and the noun sleep, I note that "went right to sleep" is well-attested, which lines up with "went right to school", "went right to the heart of the problem", and so on; whereas we don't say *"went right to talk to her", or *"went right to see what was happening", or the like.
Another piece of evidence is the related idiom "put <someone> to sleep"; put takes a prepositional phrase or the grammatical equivalent ("put <something> in the trash", "put <someone> outside", etc.), not an infinitive (*"put <someone> to be <participle>", *"put <someone> to talk to her").
Conversely, I can't find any evidence that any occurrences are using the to-infinitive to sleep; I tried a few different kinds of searches:
- "went to <adverb> sleep" — I tried various adverbs, such as just, quickly, and merely, none of which were attested; however, corresponding searches with other verbs did not get many hits, either, so the lack of them for sleep is not a very compelling argument either way.
- "went to sleep <adverb-that-works-with-sleep> — all I could think of were deep and deeply, neither of which is attested (whereas "to sleep deep" and "to sleep deeply", without the "went", are both very well attested). I think this is a somewhat stronger piece of evidence, but of course I don't have any other clear-cut verb like see or talk to compare it to.
- "went to sleep <object>" — this is a very limited test, because almost the only objects that the verb sleep ever takes are noun phrases headed by the noun sleep (e.g. "to sleep the sleep of the just"); but for what it's worth, neither "went to sleep a sleep […]" nor "went to sleep the sleep […]" is attested.
So I would tentatively suggest that "go to sleep" only ever uses the preposition to and the noun sleep, even though in the vast majority of occurrences there's no way to definitively rule out the parse with the to-infinitive to sleep.