Jo, as the questing hero, comes from humble beginnings. From poverty she rises to find her true self (in this case a writer).
The typical mentor role is played by Marmee. Through her wisdom, Jo is guided toward her quest goal.
The damsel in distress is Beth. Part of Jo's struggle is to rescue Beth from death. Though she ultimately fails (the second time), she has rescued her legacy through the adoration of a life well lived.
The warrior companion is Laurie. With his aid, including some of his own hapless adventures on his own quest, Jo grows to be what she is intended to be. (Also as the warrior, Laurie wins the princess--Amy).
The dragon is at first Mr. Laurence. Ultimately he is tamed by the princess (Beth), but until then presences a formidable obstacle, at least in Jo's eyes.
The descent into the belly of the whale would probably be her time in New York, writing thrillers. Though she is a writer, she has compromised her integrity in writing these stories, which present a particularly seamy side of life.
The magic talisman might be "The Pilgrim's Progress" (a hero tale in its own right), which guides her and in a sense gives her "protection." "Knowledge is power," and through her reading of this book, Jo gains wisdom to establish her own quest.
Another archetype is the missing father. Throughout most of the book, Mr. March is absent. Even after he comes home, he rarely makes an appearance. It is the role of Marmee that doubles as the mentor and the goddess.
Jo's ultimately becoming a writer, as well as a wife, completes her quest. She has reached the kingdom she was seeking.
(By the way, a good guide to determine archetypes is to see what story elements match up with "Star Wars." Lucas intentionally wrote the movie to include almost every archetype there is. In this case, Jo is Luke Skywalker, Laurie is Han Solo, Beth is Princess Leia, Marmee is Obi Wan, etc. "The Lord of the Rings" works almost as well, though it has some missing archetypes.)