He had trained his body to wake up at the cold sensation. He could magically warm the chain with prestidigitation, fall asleep, then wake after each hour. It was a useful skill to have as a guard. If it was his shift during the night, he wouldn't be rudely awoken by a grumpy soldier; he'd already be awake. If it wasn't his shift, he'd warm the chain again and fall right back to sleep.
Zenithral swung his legs over the bedside, stood, and strode to the window. He checked it each hour since he retired for the night, looking over the lake for signs of this supposed singing lake elf. As usual, he saw nothing but the cold lake and—a glow. It was faint, but it was there. It moved ever so slowly to the east, just outside the village. Part of Zenithral was proud that he'd gotten so far in his spontaneous investigation. The other part of him wasn't looking forward to the cold.
Zenithral left the chain on his head and warmed it, then donned his winter cloak, and warmed the rest of his clothing. He braced himself, then left the Snowdrift Inn into the icy air and towards the east where the glowing figure was headed.