Uninstall DuctTape. Delete the "Program Files/DuctTape" folder if it still exists. Install DuctTape. Those files
should be copied during installation. However, if files like steam_api.dll are missing from the SDK it will not be able to get them. If you are installing/running DuctTape on a different login, then steam and the SDK must be installed and ran at least once on that login as well. To run it as Administrator, in your normal login, navigate to "Program Files/DuctTape". Right click DuctTape.exe and select run as Administrator. steam_api.dll is one of those files it needs, so hopefully it will fix your problem.
I'm actually adding a "feature" for this that will make DuctTape download/install steam, start steam, then install and run the SDKLauncher if necessary. Also gonna have it launch itself as administrator when needed, copy the files and then continue in non-administrator mode.
The disabled "Run as Administrator" problem is caused by the link being registry generated, and not a "real" link. This is because the installer is currently a Visual Studio deployment project. It will eventually be changed to WiX, but icba right now cause I know very little WiX. If another developer would like to assist, PLEASE let me know.
EDIT: Also, this isn't the appropriate place for support requests and bug reports. Please visit the sourceforge page for DuctTape and use the trackers. Make sure to use the search function to avoid duplicating issues. https://sourceforge.net/tracker/?group_id=240814