Here's (at least some of) what Kant says about this issue, in the Transcendental Dialectic:
It must, however, appear extraordinary at first sight that the condition under which I think, and which is consequently a property of my subject, should be held to be likewise valid for every existence which thinks, and that we can presume to base upon a seemingly empirical proposition a judgement which is apodeictic and universal, to wit, that everything which thinks is constituted as the voice of my consciousness declares it to be, that is, as a self-conscious being. The cause of this belief is to be found in the fact that we necessarily attribute to things a priori all the properties which constitute conditions under which alone we can cogitate them. Now I cannot obtain the least representation of a thinking being by means of external experience, but solely through self-consciousness. Such objects are consequently nothing more than the transference of this consciousness of mine to other things which can only thus be represented as thinking beings. The proposition, "I think," is, in the present case, understood in a problematical sense, not in so far as it contains a perception of an existence (like the Cartesian "Cogito, ergo sum"), but in regard to its mere possibility—for the purpose of discovering what properties may be inferred from so simple a proposition and predicated of the subject of it.
If at the foundation of our pure rational cognition of thinking beings there lay more than the mere Cogito—if we could likewise call in aid observations on the play of our thoughts, and the thence derived natural laws of the thinking self, there would arise an empirical psychology which would be a kind of physiology of the internal sense and might possibly be capable of explaining the phenomena of that sense.
One might argue that, "Time exists," is tantamount to, "I exist," and that this "I" is thinking, insofar as thinking is conflated with the action of the internal sense; but so on the technicalities of Kantspeak, the internal sense is not the faculty of thought (because of the dynamical distinction between intuition and discursion), and so intuition of my thoughts (intuition of my very faculty of discursion) might seem as a priori as time or as empirical as the objects of time. I suspect that this matter is a moment of potential inconsistency in Kant's theory, or consistency at a very high, and not very clearly known, price.
EDIT: Consider these statements:
- The faculty of intuition is different from the faculty of discursion.
- The faculties of intuition and discursion exist.
(1) is analytic in that the difference is conceptually displayed; (2) is synthetic because it is an existence claim; but it is perhaps difficult bordering on impossible to say whether (2) is then a priori or empirical or rather something else (a combination of those two characteristics or a separate characteristic altogether). I wish I remembered which of his books it was in, but one neo-Kantian scholar, Allen Wood, defends an empiricist reading of (2) (or of claims akin to (2)), if I remember correctly; I'm not the biggest fan, by any means, of Wood (he's pretty strident betimes), but he's a solid analyst, so worth looking into.