I was in something of the same situation...my husband had been in the US with me for eight months. As we were both jobless, we got a letter from his mum stating we could live with her and that she would support us. We had to back it up with a letter proving the mortgage as well as her payslips and bank statements. They didn't look at our bank account because, frankly, they would have laughed at the big red number we had gotten to by that point!
I don't know if £2,000 will be deemed sufficient to support you until you find a job, honestly. It seems completely subjective as to the amounts which are considered acceptable. Maybe someone else will have more concrete advice; for us, using his mum's income (and she's a pensioner) and proof was enough to get us the visa with no hassles.