I have a Kenmore Elite HE3 dryer model #

Your dryer is electric. This means it needs 2 electrical phases (hot wires) and one neutral to wor. If you measure 110v across 2 phases - one of them is out.
Unplug the dryer. Find the double breaker for your dryer on the breaker panel.
Pull breaker to the off position hard (do not overdo it,as you can pull the breaker out), and then back to on position.
Check voltage between the two phases, it should read 220-240v. Check the voltage between each phase and the neutral (middle prong), should read 110-120v. If you don't get these readings - call electrician.
If you get the proper voltage, plug the dryer back in. If it does not work - Dryer needs repair. Likely a thermal fuse is blown or problem with power wiring inside.