Unfortunately, this libeay32.ddl is all over the place, including a number of places a Windows search doesn't want to show me. And, I have no idea which one to use or not to use as none seemed to have arrived with the perl module that is using it.
I didn't find the dll in any of the special locations like system32 that I have been warned about. However, I did find one in the MKS Toolkit which is always a big red flag to me. The network administrators use this crap all the time and update it whenever they like, and it seems to continually get in the way of my stuff. I moved MKS Toolkit to the end of the path, and everything seems to be working again. Maybe I broke something of their's and won't be able to log in tomorrow, but at least I'll be able to do something useful today.