How do I connect (which wires to what relay terminations) the DPDT relay in a KitchenAid Proline KES 100?

Best to use a relay , the terminals are marked L for lights, B for power connection, S for switch. Mount the relay close to the battery, run a 4mm cable with a inline fuse , from the b+ cable to the b terminal on the relay, brake into the light cable and run a wire from it to the S terminal on the relay, Connect the shadow lights to the L terminal on the relay, make sure the lights are earthed.

Electricians don't guess, they test.
Usually 'relays' used for HVAC are called contactors, for some reason.

1) 24Volt wires connect to coil. Make sure contactor coil is rated 24Volts or 24vac
Coil terminals are typically located on side of contactor.
Usually you can see small copper wires from coil that are soldered to these terminals.
Test across terminals with multimeter set to read ohms, and you will get ohm reading of coil.

2) The 120V or 240V wires connect to terminals located on ends -or- on top of contactor.
If there are three terminals, then check these terminals with multimeter.
Push down contactor to simulate the contactor turning on and off.
Hold contactor down and test terminals again.
And compare test results with same test from old contactor being replaced.

3) Open following link to see typical 24V coil contactor with DPST with 2 NO contacts
The contactor shown in diagram is similar to your contactor, except the contactor shown has 4 terminals for 120V or 240V. This is why testing contactor and comparing with old contactor is necessary.
http://waterheatertimer.org/images/ON-delay-timer-diagram2-300.jpg

Open following links to identify timer and download manual:
http://waterheatertimer.org/Intermatic-timers-and-manuals.html#GM

GM40 can be used for 120V or 240V 277V applications for single-phase and some 3-phase applications.
For 120Volt, set all 4 dip switches to ON position BEFORE applying power.
ON position is pushed to left.

120Volt cable from breaker box has black hot and white neutral wire.
Black hot typically connects to terminal L1
White neutral connects to terminal L2/N
This action will power the time clock and relays.

Other terminals on GM40 are the relay terminals, and these terminals are dry.
Dry means they receive no power when wires are connected to L1 and L2/N.

There are two independent relays and each relay has 3 terminals: NC NO and COM.
If you are controlling one 120volt Load, then use one relay.
Run a jumper wire from terminal L1 to COM terminal.
So COM terminal is connected to Hot.

Cable going to 120V Load has black and white wire.
Connect black Load wire to terminal NO (normally open).
Connect white Load wire to L2/N.

On the control: Check voltage Connector J1-3 (Black wire to # 3) to any L2 red wire you should get 220 v . If not the bi-metal thermostat is probably open. If voltage is ok check with oven on bake between P 11 Red wire to L2 red (at main connection) should read 220 volts. If not the timer is bad if so check DLB relay . Take 1 wire off switch side of relay, activate bake. Coil side should read 120 v switch should read 0 ohms.Relay ok . Check between the wire you took off and the other wire you left on and the reading should be 220 v.

Good day,
You can go here to see the basic wiring setup of your terminals for the compressor along with the relay and overload.

http://www.jimdow.com/jimdow/sales/diagrams/refcopressorterm.jpg

L1 andL2 can be interchanged, since it is alternating current.
Basically, the overload is clipped to the body of the compressor, and is wired to the top center pin (common).
The relay, is pressed onto the R & S terminals. L1 or L2 can be used to connect to the overload terminal. The unused one is connected to the R side of the relay.
In addition, there is a run capacitor that is jumpered between R & S.
Actually it is an accessory to improve the effeciency of the compressor. It would run without it.
Hope this helps..

Thanks for using Fix Ya

A three wire connection dryer cord hookup. Notice that there is a ground strap connected from the common center terminal to the frame of the terminal block. Place the center wire of the cord on the center terminal [ one with the ground ******** it ] and the other two wires will be placed on the outer terminals and it doesn't matter which wire goes right or left. Get the center wire of the cord right and you are in business.
A 4 wire dryer cord hookup. Notice that the ground strap has been removed from the terminal block frame and folded back onto the center terminal for storage. You can see the colors of the other hookups. TIP: White always to the middle.

Here is a link to a very informative site
http://www.applianceaid.com/general.html#3to4

Changing from 3 to 4 wire hookup on a electric dryer:
Power cord help for 3 & 4 prong dryer cords 
This is an example of a three wire connection dryer cord hookup. Notice that there is a ground strap connected from the common center terminal to the frame of the terminal block. Place the center wire of the cord on the center terminal [ one with the ground ******** it ] and the other two wires will be placed on the outer terminals and it doesn't matter which wire goes right or left. Get the center wire of the cord right and you are in business. 
This is an example of the 4 wire dryer cord hookup. Notice that the ground strap has been removed from the terminal block frame and folded back onto the center terminal for storage. You can see the colors of the other hookups. TIP: White always to the middle.

The wiring on these is pretty straight forward

Basically a contactor, either DPST or DPDT, depending. Input power connects here.

Compressor C terminal connects to one side. R terminal connects to the other. 1 terminal of the run capacitor connects to the terminal with the R terminal of compressor. The other side of run capacitor hooks to S terminal of compressor. One wire of condensor fan hooks with compressor R terminal, the other condensor fan lead hooks with compressor C terminal.

Only things missing from all this is a hard start kit and crankcase heater. Let me know if you want to know how to hook those up.

hopefull yyou havent disconnect old switch yet. whirpool and kit aid are same Co. same parts used. assuming the correct part was sent for the corrrect burner. diffrerent element will require different rated swties(ie. small , large, dual and so forth.), the indivudal switches are labeled, on the swithc itself ( L1,L2,H1 H2, P1 andso forth, so although the switches looke differnet they should stil have the same lettered terminal to plug into .