diligenter

Diligenter is the adverb of diligens, which is itself the present participle of diligo, "to love, to single out". The logic is worth keeping: the careful person is the one who has picked something out to care about, so diligentia is closer to devoted attention than to English "diligence" as mere grind.

The form is the point. Adjectives whose stem ends in -nt- build their adverb in -nter, not in -ente: diligenter, prudenter, constanter, libenter, sapienter. Comparative diligentius, superlative diligentissime, both regular. Do not confuse diligo, diligere, dilexi, dilectum (to esteem, love) with diligo built on lego in the sense of gathering; the Vulgate's diliges Dominum Deum tuum is the first of these.