I learned this nigun from the great-granddaughter of the Skolyer Rebbe, who learned it from her grandmother, who would sing the entire prayer sitting in front of the menorah watching the Hanukah candles burn. (It also works for counting the omer.)
According to Reb Moshe Aharon (Miles Krassen), it is attributed to the Besht (Baal Shem Tov). This is a true deveikes nigun (devekut = meditation/cleaving to God). Doing the whole song ends up being a pretty long meditation, since the Skolyer tradition is to repeat each word seven times (making 42 verses).
This recording includes the first line alone — see also the adapted version below that goes through the whole prayer. It was recorded when Hillel Lester gathered a bunch of us in a studio to sing some of our favorites together. Here's what we came up with - a rough cut but a sweet one. Starting with one voice, this unrehearsed track fans out to improvised harmonies and then back to unison, one word per verse in the traditional Skolyer manner.