You mention intuition, so you want some account, not given so far, of intuition. This term, like empiricism itself, is open to a broad variety of meanings. My own understanding of intuition is that if I know something intuitively, I know it by a direct intellectual awareness not derived from sense experience. It follows that intuition - intuitive knowledge - is not empirical knowledge. Empiricism excludes intuition.
Intuition or intuitive knowledge is not rational knowledge either. Intuition involves 'instantaneous insight' (B. Williams, Descartes, London: Routledge, 2005: 73) not preceded by rational processes.
This does not connect intuition with the sensesirrational, however. Intuition delivers truth. It is 'instantaneous insight' into the truth. Irrationalism by contrast involves - endorses or promotes - intellectual arbitrariness, private whim and wishful thinking, none of which can deliver truth, an essential element in knowledge, other than accidentally.
Whether we actually have any intuitive knowledge as defined I cannot say.
