The "rat" the lady wore in her hair was a pad of false hair - for some reason it was called a rat. Those Edwardian hairstyles were very puffy and demanded extra padding, so the lady's combs and brushes were cleaned (by her maid, of course) of any loose hair which had shed during grooming, then the loose hair was packed into a small dish and over time grew quite substantially - hence, a matching rat.

Nothing to do with the song, I know, but I thought it might clear up the line in the song, in case you thought she was infested with rodents.

Cheers
JennieG