Different Gamertag
One potential reason that a "Borderlands" character is not appearing is because it is associated with a different gamertag than the one that was recovered. In order to play local multi-player, each character must be made with a separate gamertag. It is impossible for two characters saved to the same gamertag to be used at the same time. If you regularly play split-screen multi-player, then it is possible you incorrectly remembered which gamertag the character was made with. Try logging in to any other gamertags used on your console to see if this is the issue.
"Borderlands" Save Location
Recovering a gamertag allows you to move your profile to a different Xbox 360 console without directly transferring the old console's hard drive to the new one. Some information, such as achievements, is saved online to your gamertag profile, but not everything. The characters made in "Borderlands" are saved to the hard drive of the Xbox 360. If you recover your gamertag on a different console, the save data for your characters still has not been transferred. You must transfer your gamertag and the save data from the old console to use your "Borderlands" characters on a different console.
Deleted Characters
Deleting a gamertag profile on your Xbox 360 can potentially also delete all of the save data associated with it. It is possible to delete a gamertag from a profile without deleting the save data depending on how it is deleted. It is also possible to separately delete the "Borderlands" save data. Your characters are gone completely if you delete the "Borderlands" save data directly or indirectly when deleting the profile. It does not matter how you recover the gamertag, the save data for your characters no longer exists to access.
External Storage
The Xbox 360 allows save data to be transferred to memory cards or flash drives. Similar to deleted save data, you cannot access your "Borderlands" characters if you recover your gamertag in this situation. However, your characters are recoverable in this situation. All you have to do is copy the "Borderlands" save data to the hard drive of the Xbox 360 where you recovered your gamertag. You can access your characters once the save data and gamertag are stored on the same console.