The key to understanding this poem is recognising how Dickinson is creating a contrast between the definition of being a "nobody" and a "somebody," and how she plays with those definitions to present being a "nobody" in a positive light and being a "somebody" in a negative light. In the first stanza, being a nobody is something that the speaker is very keen to keep to herself. She implores the reader to not "tell" in fear that "they" would banish her and any other "nobodies." In the second stanza, note how this negative presentation of being a "somebody" is achieved:
How dreary to be somebody!
How public like a frog
To tell one's name the livelong day
To an admiring bog!
Dickinson chooses to characterise being a "somebody" through stating that it is "dreary" and she compares this state to being like a frog who is left calling out its name all day to an audience that is no more important than an "admiring bog." The word ignorance, if it was going to be attached to this poem, would therefore be best linked to Dickinson's presentation of being a "somebody" and their activity in being so "public" that they have to repeat their name endlessly, demonstrating ignorance in the way that they fail to recognise how "dreary" and futile an activity it is to repeat their name to nobody except their surroundings.
Ignorance might be defined therefore as a state of feeling the need to announce your presence no matter whether there is anybody around to take notice. Far better, the poem suggests, to be a secretive "nobody."