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The following 2001 CLK55 AMG illuminated door sills installation was performed by and documented by Joe Navek, who has given permission for their presentation by 
step 1, Remove Passenger side stock door sill. This can be done by pulling up on the ends of the sill. If necessary, use a trim piece removal tool to assist. Be careful to prevent distorting sill.

1aa: underside of stock sill.

step 2, Open wire access panels

step 2a, Find small brown wire with blue stripe and large red 12V source as seen below. Note that you should be using the appropriate gauge size wire splicers as seen below. They can be purchased at any electronics store.
step 2aa, After finding the small brown wire with blue stripe and the large red wire. Splice together the small brown-blue with the black wire from the light kit (ground.) Next, splice together the large red wire with the red wire from the kit (12V constant.)
step 3, Neatly fold wires back into wire housing out of way.
step 4, Splice together corresponding colors from the ends of the kit's newly spliced wire to relay wires. Insulate exposed leads with black electrical tape or wire heat-shrink wrap.
step 4a, Verify proper connection by plugging in light strip.
step 4aa, Light should go out once the dome light button is depressed.
step 5, Peel off the paper on the light strip (front.)
step 5a, Center the light strip in allotted area (it may help to do this while looking from the front in order to position more accurately.) After light is centered, drill a 1 inch hole ~ 13 inches from the left side of stock door sill (see next picture for reference) for the D/S, drill hole on the RIGHT side.)) Wipe entire stock sill down with a "body prep" solvent to clean off fingerprints, grease and wax. Next, remove red 3M tape from adhesive strips and center new lighted sill over stock (run wire through your 1 inch hole first.) Use the ends of the stock sill to gauge accuracy of centered position.
step 6, Hook-up the ends of the wires and verify everything is working properly. the light strip should be on, and turn off when dome switch is depressed.
step 7, Position new sill centered in doorway and gently tap down in place with open palm of hand. wipe down with glass cleaner to remove fingerprints.
step8, Follow same procedure for driver's side, but look for the small brown wire with the gray stripe (ground) and the large 12V wire will have a purple stripe.