The medical standard for Army per AR 40-501, Standards of Medical Fitness,
for asthma is
d. Asthma (493), including reactive airway disease, exercise-induced
broncho spasm or asthmatic bronchitis, reliably diagnosed and symptomatic
after the 13th birthday, does not meet the standard. Reliable diagnostic
criteria may include any of the following elements: substantiated history of
cough, wheeze, chest tightness, and/or dyspnea that persists or recurs over
a prolonged period of time, generally more than 12 months.
My recommendation is to talk to your doctor and take medical documentation concernign your past asthma to DoDMERB exam. You have to rememeber, the DoDMERB exam is nothing but a screening in a way. In absence of any medical evidence at the time, easier for the doctor to note "history of asthams after 13, disqualified."
DoDMERB does not issue any waiver, individual SA issues waiver. For West Point, for strong candidates West Point will request waivers and depends on your condition you might get a waiver.