Although there are different techniques used to divide daylilies, the goal is to separate the fans with as little damage as possible to the fan and it's roots. A tool is needed to pry or cut apart the fans at the base or crown. Many use a screwdriver, serrated bread knife or a hand saw. Some slice between them with a shovel, while others use two screwdrivers prying apart in opposite directions.
It will be easier to see what you are doing if you cut back the foliage by at least half. This will also allow the newly divided and replanted fans to adjust more easily. Dig up the entire clump, including about 6 - 8 inches of soil surrounding the clump (many of the roots grow out from the center). Wash off as much dirt as possible. Find where the fans are connected at the crown of the plant - that is where you carefully cut or pry them apart. Some fans are not connected, you merely need to disentangle their roots. Breakage or cutting of some of the roots is inevitable, but try to preserve as much as possible. If a fan snaps off of the crown, the leaves will not grow new roots. But it is possible that the crown, if the roots are attached, will sprout new leaves.
Replant each fan at least a foot apart. Dig a hole that will accomodate the roots, and make a mound in the hole so that the crown sits on the mound, just below ground level, and the roots spread out and down from the crown.