There's no reason to delete the database, just delete the entries for Links Manager. There are six tables starting with Links. There is one entry in the configuration_group table and 18 in configuration (assuming you installed them all). You can search for the work link in each to help find them. Once all of those are deleted, try running links_setup again.
But, unless you know the database might be wrong, you should first try replacing all of your files from the contribution (make backups first so they can be restored). Does the problem still happen? If so, try the database as mentioned above. Does it still happen? If so, then you hae a problem unrelated to Lnks Manager I'm afraid.
Jack