Notice the glow of the orange flame from the ignitor but no gas ignition. If the ignitor is cycling on and of and the gas valve is not letting gas to the burner you have bad booster coils. These are the two round coils on top of the valve. When the ignitor is glowing and drawing enough amperage the coils will energize and gas will pour out to the ignitor to be ignited.
Most Whirlpool/Kenmore style dryers, here is how to access and repair your dryer.
-Remove the lint filter
-Remove two screws under the dryers lint filter (On, lint filter on top models)
-Insert putty knife where front meets dryers top, about two inches from both corners push in and top will lift.(Lint filter in door model)
-Remove two 5/16 screws located on upper front inside panel
-Lift front panel up and out of lower clips
-Unhook dryers door switch
-Reach in and loosen and remove belt
-Pull the drum up and off the rollers
Now you are inside the dryer and are able to see the booster coils, on top of the gas valve they are mounted with a solid metal housing remove the screw and lift the cover off. Unhook the wires from the coils and replace, Install the new set. Looking on the top of the coils you will notice a small plastic abrasion this must line up with the metal cover when getting reinstalled.
Put the unit back together and listen for the sound of the gas lighting. If your dryers model has an access cover, you are able to watch from a distance the flame light. If you are sure the gas is turned on and you have good pressure replacing the booster coils will take care of the problem. You can ohm the coils out if a multimeter is available,not everyone has one laying around to test voltage.