My asthma causes a cough like this every time I get a cold. It will linger for literally months if I don't fix it... like three or more. So, whenever I get a cold, I take hydrocodone cough syrup and use my inhaler. The dry cough I get is basically irritated lungs- I cough, it irritates my lungs, I cough because my lungs are irritated, which irritates my lungs, so I cough more... and so on. Endlessly. It's much worse at night because laying down irritates lungs too.
I'll also get these coughing fits after I have a cold- again, for months afterwards- where I will choke, cough every last bit of air out of my lungs, and then keep coughing till I throw up sometimes. Awesome. Made for fun times when I was a teenager, before I'd realized how to fix it. (I still remember teachers getting angry at how disruptively annoying my coughing was - one teacher pelted me with cough drops whenever I'd start hacking.)
Go to doctor. Get codeine or hydrocodone cough syrup; personally I find the codeine variety doesn't work as well as the hydrocodone kind. Take it at night, it is the BEST fix for coughing ever. Of course don't drive when taking it or try to do anything seriously complex! Usually I just need it at night. If you're not having the coughing fits like I described you've probably just irritated your lungs, no asthma as a bonus to make it worse. Essentially, the inhaler is for the coughing fits which are asthma related, and the hydrocodone cough syrup is for the dry cough at night... so if you're just having dry cough at night, the syrup should do the trick.
My sister just got a terrible cold that gave her a month long cough just like you described, and I told ther the same thing- she couldn't believe how fast her cough went away after she got some prescription cough syrup.
(Not to advocate off-label use of a prescription, but just in case you have some old Vicodin laying around from wisdom teeth getting pulled or something, that'll do the exact same thing as the cough syrup. The active ingredient is the same, it's just pill form with Tylenol rather than a syrup. If I remember, they add some sort of expectorant to the cough syrup, but if you have a dry cough it's not the important ingredient.)