The sudden shutdown may have corrupted the file system. Go to Computer and right-click the C: drive the select Properties. Under the Tools tab for Error checking hit the Check Now button.
Leave just the first box ticked and then hit the the Start button. It will tell you it has to schedule for next reboot. Say OK and reboot.
See if it runs any better. If not repeat the steps but this time tick both boxes before scheduling the error checking. (this adds a considerable amount of time to the process (up to an hour) so restart only when you have time.
You may not be able to, the battery is there to buffer the system from the main power source. A spike in power as in a sudden drop or surge can damage your computer. Take it in and and have diagnosed for repair.
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