How old is this cow?
Have you noticed any deformation of the outside of her nose or skull?
Are you seeing any diarrhea?
Is it one nostril or both?
Has she been given any oral tubes lately? Like a calcium gel?
Does she fight with other cattle?
How long has this being going on (the bleeding)?
Have you noticed any coughing?
Do you have any history of more recent respiratory disease in your herd?
What breed is she?
Unfortunately you seemed to have ruled out the easy, good reasons.
Trauma is the primary reason for bleeding from both nostrils. With the loss in condition as well as the bleeding you must think of some disease process.
The the most common reason then are lung abscesses. These usually start in the liver and spread to the lungs. The liver is also responsible for making some of the blood clotting factors. So once abscesses in the lungs rupture they will continue to bleed and not clot very well.
Since she has been loosing condition and you are not seeing symptoms in any other animals then toxic plants (that interfere with blood clotting would be much less likely).
For now you can try an antibiotic that penetrates lung tissue well. Penicillin, Naxcel or a sulfa antibiotic. Most of the time these animals continue to decline and do not recover. Liver abscesses can come from blood borne bacteria or parasite migration.
If I can answer anything else please let me know.