Synthetic

Not sure either, but my first guess is she might have dried up membranes in her nose from being sick and that she is dehydrated.

My first thought was winter dryness in the air too.

I would try to get more water in her and see if this helps. Sick cats like sick people can get dehydrated as hell and the nose can get dried up inside too from colds.

I have to run to bed but I will check back tomorrow night on this and hope she stops bleeding.

Never had a cat with a bloody nose. It was on her left side...she is so white too, the blood was easy to see, ugh....

Happened last night. I'm doubling down on all the antibiotics. If she isn't well by Friday, we're going to the vet when I can get someone to take the drive out for me. 

Get some more water in her and see if that helps. She might be dehdrated with a dried up nose from being sick, hopefully this is nothing serious.