Apparently the issue is with PHP not being able to handle a computer name with underscores in it (which is permitted by Windows networking).
I took underscores out of the name as well as any other punctuation marks, rebooted, attempted a reinstallation of the wiki, and everything went well (this was with a BitNami stack, but I'm going to uninstall it and try to do a straight IIS installation next).
To see if this is the problem, do the following against a malfunctioning wiki login.
1. Open up a browser on the machine running IIS.
2. Instead of http://<ComputerName>/<sitename>/doku.php, use
http://localhost/<sitename>/doku.php in the address bar.
3. Click Login and enter the admin credentials.
4. If you can now see the admin button you have successfully logged in, and your IIS server likely has a name that PHP is choking on.
In this case (and if it is an option in your IT environment) change the name of the computer to one without punctuation, reboot, and reattempt the login using http://<ComputerName>/<sitename>/doku.php. It should work now.