Lang/langt/lange is an adjective or may – in the case of langt be an adverb of distance, længe is a time adverb:
Det var en lang nat for en lang dreng i en kort seng - It was a long night for a tall boy in a short bed (adjective)
De havde kørt langt – They had driven a long way. (adverb of distance)
Jeg har længe gerne villet prøve NOMA - I have long wanted to try out NOMA (time adverb)
Lang goes together with a noun/pronoun - and conjugate accordingly: lang, langt, lange, langt may describe a distance - whereas længe stands alone or goes with a verb.