If the italicized sentence is <em>how eager she was to do her best, </em>then the answer would be an adverb clause.
Because the whole clause acts as an adverb.
I dont know if you got the answer already but I think its the first one third and last! I dont know if its right.
According to Liza, "ladies" are cleaner than commoners because poor people do not have the luxuries that make bathing pleasant.
Liza says: "I tell you, it's easy to clean up here. Hot and cold water on tap, just as much as you like, there is. Woolly towels, there is; and a towel horse so hot, it burns your fingers. Soft brushes to scrub yourself, and a wooden bowl of soap smelling like primroses. Now I know why ladies is so clean. Washing's a treat for them. Wish they saw what it is for the like of me!"
This quote clearly states that, before the bath she was given by Higgins' housekeeper, bathing for her was not such a pleasant activity (probably she bathed with cold water and unscented sopa, if any.)
