As for choosing a place to eat at I would recommend the restaurantguru.com website. It has convenient search options (by address, nearest city attractions, venue type, cuisines), moreover it has visitors opinions gathered from all popular web services.
As for the two, you are asking about, I would choose none of them, for the atmosphere and decor first of all. But that's only my opinion... Probably, it's because I'am not a fan of Michelin restaurants, and their food... I prefer smth cosy, warm small places with intimate atmosphere, and with good portion of food on my husband's plate)))
As for a place to stay in, I used to search for a flat to rent on gumtree.com. Btw, as far as I know, the rent payment in Islington is higher than in Battersea.
Good stay and happy eating!)))