As an asthmatic - anything you do to increase lung capacity (regardless of what it might do to your throat to the positive - that was a new one on me) is good for your asthma.

ANY wind instrument, particularly the larger ones, will work for this. I'm also with Callistabee on this one - practice valves, etc. are just as handy and may be less embarrassing / costly / bulky.

Going straight to the didgeridoo may be overkill and frustrating. While I haven't played one, I understand that they're a VERY LARGE air column, and she might be less irritated starting with something a bit smaller and working her way up.