Several programs have errors and won't start up after turning off my computer while it was updating after I shut it down for the night. However, the next morning the updates (apparently) were still being installed, but I needed the PC so I hard booted it. Then the programs failed. Is there a way I can complete the installation and repair the errors?

Try going into the add/remove programs and remove the recent updates or download them yourself from windows update and reinstall them by hand.