The nicely detailed history you have provided does, indeed, suggest the diagnosis of, Cough-variant Asthma but that diagnosis is inconsistent with the total lack of response (I believe that is what you indicate) to Asmanex, Singulair and albuterol. You note that, “it is rare for me to get more than one cold a year”. That is consistent with a seasonal-cause, be it allergy or infection but would not account for the 3-5 month persistence of cough. Nevertheless, it would be wise for you and your doctors should give careful consideration to potential environmental exposure causes, be the exposure outdoors, at work or in your home.
Two other possibilities come to mind: 1) that you suffer from type of immunodeficiency state that precludes your body from mustering a good response to whatever organism is the cause of your “cold” and, 2) that you have an underlying (non-asthmatic) physical disease of the lungs such as, a) the so-called right middle lobe syndrome) or, b) a condition called bronchiectasis, a physical disorder of the bronchial tubes that predisposes one to lung infection and then is associated with very slow clearing of the infection. Bronchiectasis can be diffuse (throughout one or both lungs) or localized and it may be necessary for you to have a CT Scan of the lungs to make this, or similar diagnoses.
As you know, I am sure, there are a great number of causes of chronic and/or recurrent cough, the causes residing in the upper (nose, throat, sinuses) or lower respiratory tract (the lungs) or even in the gastrointestinal system, most commonly with gastro-esophageal reflux disease (GERD), this latter unlikely in your case, given the recurrent, long periods of relief that have characterized your illness.
Finally, many academic medical centers have Cough Clinics and you might wish to inquire about this in your geographic area. The most renowned physician in the area of Cough is Dr. Richard Irwin. You might want to arrange to see him or, at least, contact him for a recommendation of a cough specialist closer to your home.
Good luck