breath control question

I recently picked up a Waltons Irish whistle (D) and am teaching myself to play it.
I am finding that it takes VERY little air flow to play it

(far less than the clarinet I played, poorly, as a kid) and as such am having trouble with breath control. I find I end up not taking breaths because I have not used the air in my lungs yet, and longer notes tend to waver as it is hard to maintain a constant flow when blowing so little air.
Anyone have ideas and/or suggestions as to what I may be doing wrong or what I can do to make this better?