Congratulations J. on your sweet baby girl! It sounds like you two got off to a fantastic start...what a blessing!
I have had 7 wonderful babies, and I can remember 15 years ago having this exact same issue you mention with my 1st baby! I remember those same feelings of frustration as you describe. Funny, none of my other babies have done that since. They're all so different.
For me, three things worked. I couldn't stand pumps so I stayed away from those. :)
1st, once baby sucked a few seconds my let down would automatically follow (eventually)whether or not she continued to suck. Sometimes there was a 20 second delay and sometimes a 60 second delay. When I took a deep breath and forced myself to relax and think peaceful thoughts it was usually 20 seconds, but if I was tense it was more like 60. So if baby sucked a little & triggered let down on the left side, I would switch to the right and wait til I felt let down on the left, then switch back. That would keep her distracted and sucking at the same time. If I thought of it ahead of time, I could even stimulate my own let-down right before beginning to nurse by rubbing my nipple back and forth between my thumb and index finger.
2nd, I later figured out that my baby became easily frustrated at the breast when she was starving, but was less agitated if she was less hungry. That's about the time I read Dr. Sears' lifechanging "The Baby Book" and changed my attitude about frequency of nursing. (The more the better!) By nursing every 2 hours (or more if needed) and always upon waking and right before going to sleep (which should never be more than 2 hours) my baby was eating BEFORE she was starving and therefore was much more relaxed and patient. What a relief that was! This new rhythym also brought improvements all around: in sleep, behavior/temperment, growth/development, etc. I had a happier baby and I was a happier mommy.
3rd, and also as a result of reading Dr. Sears, I got permission to follow my instincts and bring baby to bed with me. (I never felt like the sudden forced separation between a mother and infant was natural, so soon after the closeness of womblife and all). Now imagine what co-sleeping does to this whole problem...baby is able to nurse at will during the night without mommy even having to wake-up (Daddy was thrilled to see his girls VERY well-rested!) Since baby is half-asleep while she's nursing, she doesn't notice the delay in let down...and since Mommy's completely (or nearly) asleep, she's relaxed and let down is very quick. So baby gets lots of good nursing time without any frustration at all. Then in the daytime she is more content and used to nursing without crying. She naturally calms down and learns to enjoy non-nutritive sucking as much as "eating," and the problem is solved.
So needless to say, as I read, prayed, and learned, my problem went away very quickly. In fact, once I figured out the joys of "natural mothering" all my problems went away and mothering became very easy and lots of fun. I just had to let go of a lot of preconceived notions and myths that had been passed down to me. And here I am, 6 babies later (with a 9 month old)having the time of my life! And those days of trying to calm Baby #1 while waiting for my milk to let down seem like a lifetime ago. But they're precious memories all the same. Because with your first child (or your first girl!), everything is so fresh and unexpected and amazing and unforgettable. Enjoy!
God bless you and your sweet family,
L.