If you had a widget initially, and you were getting an eval error, that means either your custom PHP code within the widget was incorrect, or you had an incorrect custom condition entered in the plugin's settings section. If you decided to fix it by altering plugin's source code and DB records, and your widget completely disappeared, I don't know what to recommend you beside wiping out that DB record completely, removing the plugin and re-installing it again. If I were in this mess, that's what I'd do. I can't magically see what could be wrong with the plugin, especially knowing that you made manual alterations in multiple places. I know you said that you "removed" the code you entered before, but the plugin is not functioning as it did before, which means it's not in the exact state as it was before your alterations.
When re-installing the plugin, do the following:
1. remove the plugin completely from the site - delete it via FTP.
2. edit your DB - find "atw" option name in wp_options table and remove that record.
3. re-install the plugin.
4. Make sure ATW settings are functioning under the "Settings" section of your WP Dashboard (don't change anything yet).
5. Go into Widgets and see if the ATW widget is available.
If 4 and 5 are there - then you successfully re-installed the plugin. Don't edit the source code and don't alter serialized data manually in the DB.