Hi. Kings Cross isn't really a residential area. Well people do live there, but it's not somewhere I'd want to live with kids. Rather than looking necessarily at places in close proximity, you should be looking at places which you can get to Kings Cross easily. You're in luck as Kings Cross has lots of tube lines running to it, so it opens up lots of places for you.
Have a look at the London Tube map
I agree with Islington as a good bet. The tube stops you have in Islington are (in order of my personal preference) Angel (Northern Line), Highbury & Islington (Victoria Line), Caledonian Rd and Holloway (Picadilly Line). Holloway is possibly a bit rough around the edges (depends what you're after!)
Looking a bit more at the tube lines, firstly the Northern line (black)
There are lots of nice places to live on this line.
The following are really desirable but probably out of your price range, but worth a look just in case!
Camden Town
Chalk Farm
Belsize Park
Hampsted (seriously posh!)
Highgate
These might be slightly more affordable, also worth a look:
Tuftnell Park
Archway
Kentish Town
Victoria Line (light blue). Going north after Islington you have:
Finsbury Park: I'm not sure I would advise Finsbury Park as a first place to linve in London. Some people like it but I'm not keen personally. Too busy. Too many dodgy characters hanging around at night IME.
Tottenham / Seven Sisters - Avoid Avoid Avoid. Tottenham is a miserable, horrible place IMO!
Blackhorse Road / Walthamstow - these used to be quite unfashionable, but lots of families with young children have been moving out that way recently, because houses are more affordable. I know some people are very keen on it. Your money will go further there, could well be worth a look.
Another nice place to live is Alexandra Palace. (It has a train line into Kings Cross - see this map for London Rail and tube connections
Some friends of mine have just moved there with their LO and they love it.