Alright, there's a bit to unpick here. The first thing I'd say, is that it's not worthy of a 2 - I mean, it's not a perfect essay, but 2 is really, really harsh here. Perhaps take that as a compliment here though, your teacher knows you can do better.
In terms of the essay itself, there are two main problems I can identify: it's a bit vague, and it doesn't answer the question. By vague I mean you paint your point in broad strokes and leave out the bits like language analysis (particularly single world analysis) which is what really will get you the bulk of your marks in essay. Essentially, it feels as if you could have written about one or two less things and developed the rest much more - whenever I feel as if a good point is starting to emerge, the paragraph just seems to finish.
Then, my second point, it doesn't answer the question. You were supposed to write an essay on how Russell presents Mrs Lyons as a lonely and unhappy character, but you ended up writing an essay on how Russell presents Mrs Lyons, giving your own interpretation. You can see this most clearly by looking at your conclusion - which I don't think is necessary even, although I know this divides opinion - where you say that 'Russell's intention was to show that sometimes lower class individuals can be the most genuine.' That does not tell me that Mrs Lyons is a lonely and unhappy character. You could certainly have mode a point of the juxtaposition between Mrs Johnstone and Mrs Lyons, but you would need to explain how that accentuates Mrs Lyons' loneliness and unhappiness - that could be worth a level 9 with proper development.
Although it doesn't work for everyone, I think you really would benefit from following the PEAL (or rather PEAEAEAL) structure in order to make sure that you are continuously answering the question and to keep your writing relevant and not vague with clear and concise points.