Here are your parts
Replacement parts for WHIRLPOOL LER4634PQ0 29" Electric Dryer | AppliancePartsPros.com

Here is the wiring diagram
http://www.servicematters.com/docs/wiring/Wiring%20Sheet%20-%208528187.pdf

Is the interior light coming on when the door is opened.
If not you may have a loose connection in the wall receptacle, the unit's terminal strip or the line cord itself.

Try turning the timer to different positions could be the timer contacts are not making reliably.

If still nothing the easiest/safest way to find the cause is to use a meter and check the components while the unit is unplugged.
Unplug it and set it to mid cycle timed dry and then check the cmponents for continuity.

The start circuit is fairly simple.
NEUTRAL , DOOR SWITCH , PUSH TO START SWITCH , %m centrifugal switch , MAIN and START motor windings , THERMAL FUSE , BU/BK timer contacts , L1

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

There is a good STICKY at the start of this forum about it's use.