It sounds like you may have suffered some hard drive issues from the power failure.
First, you should run chkdsk on the hard drive. Follow the Graphical Interface method instructions here to run chkdsk. At the screen where it asks what optons to check / fix, select both options.
Accept the offer to run chkdsk on the next reboot and then reboot. chkdsk will run before Windows loads. Be patient because this could take a while, depending on the size of the drive, number of errors, etc.
When that's done and it's back to the desktop, see what problems still exist..