Product review (5 stars): Replacement handle for Kenmore Elite HE3
This handle is an upgraded version of the original Kenmore handle.  It seems thicker in the section that failed with the OEM handle and has non-slip ribbing on the on underside, both of which seem like an improvement over the original.

The only thing I had to figure out was the technique to get at the screws.  Here's the procedure I followed:
There are ten Phillips screws on the inside of the dryer door. To get at the screws holding the door handle in place, you have to remove the face plate on the door front. To do this, only six of the ten Phillips screws need to be removed. There is no need to remove any of the Torex screws.

Remove the two Phillips screws at the top and the two Phillips screws at the bottom (note: bottom screws are hidden under the ledge of the door window). You are now left with six Phillips screws - three on each side. Now remove the middle Phillips screw on each side of the door. The front face plate on the door can now be removed to get access to the screws holding the door handle in place.

Remove the two screws holding the broken handle in place and remove the broken handle. Screw the new handle in place. Now reverse the procedure above to reattach the face plate. Since the six screws that were removed are all identical, you need not worry about which screw goes where.

Notes:
1: Of the remaining four Phillips screws that you need not remove, two attached the door hinge to the door and the other two just fill a hole. The two "filled holes" is where the door hinge would be attached should you wish to reverse the direction of the door swing.
2: The replacement door handle is the same part regardless of the door swing direction.