No the flex cable does not include that part. Here is the lock button, iPhone 4 and 4S Power and Lock Button. I would say you don't have to replace any parts, rather that the parts involved in the lock button are misaligned... i would carefully take apart your phone and make sure everything is in the right place. For some reason the actual metal button that you normally push on is assembled in two pieces, there is the main part (that you see), and a thin metal plate that is located on the bottom side of the lock button (that as far as i know only serves the purpose of taking up space). If this plate is missing are not set correctly in the main lock button piece it will cause problems. I would say this is most likely what is causing your problem. if this is not the case it could be the the flex cable witch may or may not need to be replaced. Either way opening up your phone is probably the best way of knowing what is actually going on, just be careful and follow the ifixit guide.
I just ran in to this problem twice just recently and it turned out to be in both cases that the tiny black spacer (really is quite small, take a look at a picture of the power button flex cable in the middle of of the actuall power button part you will see a tiny black dot) was missing, I replaced the yellow tape that this black dot was attached to but if you found something that is solid enough and is the right size you could probably just glue it right to the center of the power button part of the flex cable.