What's a lime kiln without an exhaust stack? Most stacks seemed to be circular in cross section, but the parts on hand were designed for HVAC and are rectangular. So, pulling out a few of these "printed" parts gave:
A little glue and some dry brush "rusting" and you get
Even included something that might have contained a blower or exhaust fan!
On the end of the kiln, it looked like this:
