Go to services (windows key-> search for services, or open control panel and search for services in the search box in the top right hand corner)
Find "Cygwin-sshd" in the list of services. Double click on the service and chose the "Logon" tab.
In the box next to "This account" you will see something like the following:
hostname/cyg_server
where hostname is your computer's name and cyg_server is the user created to start the service.
Replace the name of computer with a dot (.) so it looks like this:
./cyg_server
Click ok and then try to start the service.
Hat tip to Konsto in the comments at this page for finding this: http://izadpanah.blogspot.com/2010/01/cygwin-sshd-service-not-starting-on.html
(I realize I'm answering my own question, but this was very annoying and I wanted to put it somewhere so others could find it.)