This is an interesting assignment, and can be used not only to show knowledge of crucial chapters in Jane Eyre but also of the period as a whole.
First, you need to invent a fictional occasion for writing – perhaps an object Jane thinks she might have left behind while packing – because there is nothing in the text that would suggest such a letter likely, nor is there a suggestion that Jane had regular correspondence with Bessie.
Next, think about what sort of details would be included. Would Jane reveal the complex emotions the narrator tells us about in a letter to Bessie or would she simply recount the basic events and situation, e.g. that Adele, her pupil, is a young French girl, the size of the household, etc. For Rochester, the size of his fortune, his appearance, and a small amount of biographical information might be appropriate.
You might also want to look at letter-writing manuals of the period such as the Ladies Complete Letter Writer to get a sense of the appropriate generic conventions.