Answer:
There are three purposes for Jane Austen to reveal the abovementioned details:
Firstly, to show how different is Miss Bingley to Miss Elizabeth, in literature we call this "foil characters", that is that one character is the opposite of the other, or it serves to show a contranst. On this case, Miss Bingley makes very silly and uneccesary comments about Darcy's letter, whereas Elizabeth later makes much more interesting comments that grab Darcy's attention. Miss Bingley bothers Darcy, while Lizzy engages him in a conversation. Miss Bingley and Elizabeth Bennet belong to different social classes, Miss Bingley is VERY rich, Lizzie is rather poor. However, it is Elizabeth who is able to catch Darcy's attention with her intelligence and witty language. Miss Bingley is at a loss, she can only comment on superficial facts such as "handwriting, eveness of lines or length of the letter".
Secondly, Austen reveals all these details to show how much Miss Bingley likes Mr Darcy. So, Elizabeth appears to have some competition, although after they both speak, it is clear that Darcy has started falling for Elizabeth.
Finally, the purpose that Austen has is to make fun of the situation. In fact, this is an example of "satire"; that is, focusing on a specific topic to make fun of it. In this case, the whole situation is very funny because Miss Bingly wants to engage Darcy in conversation, but Darcy wants to write his letter, but she continues to interrupt him every two words, and with details that are not important at all.