I used to work around the corner from the HIX in Southwark. It's a safe area, but is very urban and commercial (by that I mean it's mostly offices and there is lots of concrete and not much grass). For dining, there are a few restaurants on Southwark Street, but more options are available on The Cut, which is about a 5 minute walk away. You are very close to the Tate Modern museum, and the river (where you can see Shakespeare's Globe Theatre), and as 13901 has already mentioned, Borough Market. Imperial War Museum isn't too far away (15-20 minute walk I would guess).

I used to live in King's Cross about 10 years ago, and it was much improved over it's past, and has only gotten better in the past 10 years. Even back then, though, Bloomsbury was always a bit nicer and quieter. You have more green space, which might be good with a 2 year old. British Museum is close. Other than that, you'll probably have to take transit to most other tourist sites.

I'd probably go for Bloomsbury over Southwark just for the green space, but think either would be fine.