I’m struggling with six 11.5-watt LED cans wired in series. With a 75-watt incandescent at the end, the other five seem to function properly, though the incandescent is dull. With an LED in its place, it gets very bright and the other five LEDs get dull. And I can modify the intensity of the five LEDs by varying the wattage of the incandescent bulb. What’s going on here and more importantly, how do I fix it?
Household lighting is not wired 'in series'. Ever. At least not if you expect it to work.
Each individual light fixture should be wired in parallel with all the others in the circuit so that they all 'see' the same voltage when the circuit is switched on.
However, from your description, it seems that you have something like this:
You fix this by disconnecting your new LAMP6, reconnecting the switched hot back where it was before, and then connecting your new LAMP6 between this switched hot and neutral like all the others are.