Here are your parts
Parts for Whirlpool LER8648PW0 Dryer - AppliancePartsPros.com

See the attachment for the wiring diagram.

I would start by checking the heating element with a meter.

Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 8 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and touched the case. This could cause what you describe and now the element is burned out.

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.
4. When you start always short the meter leads together. This will tell you that the meter is working and if there is any 0 offset.

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

[COLOR="Blue"] burned out in one corner. Do you know why this happens? I mean it's only copper connectors inside; why would it burn out all of a sudden?[/COLOR]
It probably did not burnout all of a sudden, it probably took some time.
What can happen is that the contacts can get dirty/pit over time. This pitting causes there to be some resistance between the contacts. When current passes through the contacts, heat is generated. This further degrades the contacts and so on and so on till the contacts fail.

[COLOR="Blue"]Do you know if it would be possible to just replace the burned out metal parts on the inside?[/COLOR]
I do not know of ant place that sell the timer's inmternal parts.