Your likely failure is that you have weak gas valve coils. The theory of operation for the gas valve assembly is shown in the image below. According to your description, the gas valve coils are strong enough to open the gas valves for ignition from a cold start. During subsequent ignition attempts after they are heated up, the coils are too weak to open the valves. I recommend that you replace the gas valve coils to fix this problem. Almost all of the Kenmore Series 80 dryers will use the gas valve coil kit that is part number 279834. You can order that kit from the
website. Here is a direct link for that kit:
Gas Valve Coil Kit
. If your Series 80 dryer has the lint screen that pulls out of the top panel, you can use the procedure in this video to replace the gas valve coils:
29" Dryer Gas Valve Coil Replacement
. If the lint screen is at the bottom of the opening to the dryer drum, you can use the procedure in this video:
27" Dryer Gas Valve Coil Replacement
Be sure that you
unplug the dryer
and shut off the gas supply before accessing internal components.
This information should help you repair this heating failure in your dryer. If you need more assistance, reply with additional details and the full model number of your Kenmore 80 Series dryer.
If you do not feel confident repairing this problem yourself, then you can have it repaired at your home by a Sears technician. Here is a link for the website:
Sears Home Services