It is correct to use a single thread for piecing and quilting.
Are the broken stitches on the piecing in your quilt? Or your sister's top? In either case, it is very easy to repair this type of problem by hand. You simply hide the thread knot underneath a piece, then use a "ladder" stitch to make the repair stitches invisible. There are probably videos online on how to do the ladder stitch, or websites with pictorial tutorials. It's easy to do; you just have to know how to do it.
Edit: I did a quick search on Youtube and found this video tute:
http://www.youtube.com/watch?v=5CjOXMuTFXo