Uneven LED Lighting Wired in Series
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.  







simulate this circuit &ndash; Schematic created using CircuitLab

However, from your description, it seems that you have something like this:







simulate this circuit

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.