Hi, and welcome to the QB!
What can you tell us about the fabric and batting you used? I'm surprised your needles are breaking so I'm a bit at a loss. I've never tied a quilt before but I'm currently tying buttons onto a wall quilt (oh God don't ask, really don't) and I'm using large embroidery needles and 2ply DMC floss, and it's going fine.
Are you "stab" stitching straight down through all the layers? Are you trying to rock the needle, or coming back up in a separate motion?