Take it slowly. If you can walk 10k then you can take part in something like the BUPA 10k - it's officially a run but the slowest person finished in 2 hours 12 (WAY slower than Kylie

) which isn't far off walking the whole way - the same for a half marathon I did recently, the last finisher took over 4 hours.
But you won't be walking the whole way (and you won't be finishing last either), try to build up your running from 30 seconds (or less) at a time using a programme like couch to 5k (google it) or one of the many plans out there (e.g. the Cancer Research website/BUPA 10k website etc etc). They're all pretty similar. Don't be afraid to repeat days or weeks, and don't try to do your running intervals too fast. You need to build up endurance and speed, but you don't need to do it both at the same time.
Don't be afraid of walk breaks, they can really help your training, and make sure you pay attention to little niggles before they become injuries.
And ask for help whenever you need it