At times I've had bleeding too for different reasons. Recently, I've been going through a period of high inflammation after exposure to smoke. I had a week of bleeding, the kind where you break a blood vessel and it keeps breaking causing mouthfuls of blood (yech). I hate, hate, hate that.
I had to change my entire routine to get it to stop. I had to stop all supplements to avoid blood thinners. Especially things like fish oil, garlic, or Vitamin E. Then the doctor changed a lot of meds like inhaled powders that could be irritating like Spiriva... and take Ipratropium bromide in a nebulizer instead.
I had to stop my inhaled steroid. Stop the Flutter which was making it worse which also meant I couldn't get good mucus clearance. I even had to stop my favorite thing: Hypertonic Saline and use a .09% saline instead. Then I kept increasing the percentage of saline (adding drops from a 10% vial)as I got better. The bleeding did stop. Although from not being able to cough out stuff, I had other complications, but the blood stopped.
I might mention that she had to give me 2 cortisone shots because without the inhaled steroid, my asthma got worse as I became more and more short of breath again trapping mucus.
I asked the doctor if I could take Vitamin K or Lysine and she said that it would not help me. So I asked to do blood tests measuring my K levels and my blood clotting time. All normal. However, the doctor said that when she looked at my throat, she could see numerous bright red capillaries. She said that inflammation causes the growth of additional tiny vessels. And she said if she were to do a bronch and look at my lungs, they would look like my throat. You'd see tiny capillaries about to burst and bleed. Inside my nose, it looked like that too. No need to do a bronch. She told me not to be shocked if I started bleeding again, but I was dilligent about trying to reduce inflammation. At her suggestion, I went to a beach with incredibly clean air and just took deep breaths. I did this for 4 days. Came feeling so much better. At home I put my air filter on high and just stayed in as much as possible.
Cold packs and drinking cold water as JazzysMom mentioned during the periods of bleeding. No hot liquids at all.
If the bleeding is from inflammation and not infection, the best way to stop it is by reducing inflammation through addressing the astham, allergy, irritant component. Are your doctors treating the asthma with medication? Medications can only do so much anyway. You have to change your environment to stop the triggers that cause these high levels of inflammation.