It won't actually cure insomnia, but it will give you a better reason for sitting up than "it's too cold to sleep." After all, it will require nearly a full box of tissues, or several trips to the sink to rinse the damp washcloth. It takes a while for the bleeding to stop (one may also be tempted, on a miserable January night, to simply put one's nose just outside the door, where the sub-zero temperature and wind chill could serve as well as an ice pack to slow the flow, but don't give in to that, unless absolutely necessary. It will attract the attention of the neighborhood raccoons and opossums). Half the night will be gone, and you won't have even noticed its passage.
On the plus side, if you're like me, you'll have warmed up the atmosphere fairly well with a spectacular array of good old monosyllabic terms the use of which your mother might not approve. A cold, wintry bedroom does shift rapidly under those conditions.
And, you have the added bonus of seeing the two cats fly off the bed to end up crouched in opposite corners of the room, staring at each other, one looking mightily miffed, the other smug -- as if to say, at least now you can't get those chin skritches you were working toward...
Of course, you know you're looking at a true cat person when the cats are allowed back onto the bed, snuggling on the pillow, within a half hour after the final drop of blood falls -- and there's no complaint about little ones snoring or whiskers tickling that damaged nose.