Ditto to all pp's. As a mom who wound up with supply problems after the first 5 days, it is SOO important that you work to get a full feeding in her.
Here are some things I learned from my lactation consultant (and from experience):
1) The motion of sucking is sedating to babies, and the Oxytocin in your milk is a sleeping pill. They are almost bound to fall asleep at the breast, which (as you have found) leads to a cycle of incomplete feeds and fitful sleep.
The oxytocin takes about 6 to 10 minutes to clear the bloodstream, so a good way to wake her up during a feed is to just lay her down for a few minutes. It feels less frustrating if you think of it as a technique to get a full feed, rather than "oh my lord, why won't this baby stay asleep more than 10 minutes?" LOL
2) The "baby doll" technique is one I picked up from a LLL site - support her head and neck in one hand, and cradle her legs on your lap with the other, and very gently sit her up and lay her back one or two times - like one of those baby dolls whose eyes open and close. Her eyes will open just like the doll and she will rouse up for a little while longer.
3) Talk and sing, use her name!
4) Find a good "wakey-up spot" to caress - Some people advocate tickling the feet, which seems kind of mean, but I found that gently massaging my lo's hands or ears would keep her roused up just enough to eat. I guess it was mildly annoying (but not nearly as annoying as being hungry!) You do want to be careful about touching the back of her head, as this can trigger a reflex to pull back off the breast.
5) Strip her to her diaper and cool her off a little - I would blow on my lo's face or head, and sometimes dab her face with a cool (not cold) damp washcloth. Of course, my apartment in the winter has steam heat and it was about 6 million degrees farenheit, so it was not as mean as if it were chilly in the room.
6) Seems counterintuitive, but break the latch! My lo would always go into a sucking frenzy when I reached down to pop her off. Try popping her off and on to get her going.
HTH, and congratulations, blessings on your and your family!