HarryMann said, "No problem, miniature lightweight LED ribbons/tapes are now being used in many applications, the batteries being small enough to be encompassed within the string I should think... go on, give it a go"
Why not get a small little LED flashlight and shine it at the window when you want to see the string. It should reflect back quite nicely. Shine the light mostly upwards at the string to avoid direct flashlight reflections into your eyes.
Now, what kind of chewing gum will you use to affix the front edge of the string to the window? I suspect common transparent tape would not do to well, nor would common duct tape. Umpty degrees below zero is not compatible with most adhesives I know that won't become a permanent part of the window.