Cruisemum, have you thought about run/walk training for now?
Set aside 20-30 mins, then run for 2 mins, walk for 2 mins, run for 2 mins, walk for 2 etc etc
As you get fitter, you can up the running bit to 3 mins, then 4, then 5, while keeping the walking at 2 mins
Eventually (and it will be sooner than you think!) you'll be able to run continously for 20 mins
Then you just need to extend that time gradually (no more than 10% at a time) until you can run for about an hour, at which point a 10km race might be a nice idea...
If you can get out 2-3 times a week, you'll be able to go for running for 3 mins to running for an hour in around 2 months.
If you are anything like me, you'll be amazed at how much better you sleep and also how much more energy you have in the day.
The runnersworld.co.uk website has got lots of great training plans if you'd rather have something structured to follow.
And I second the OP who said get some proper kit. A decent pair of trainers (proper running ones, not 'fashion' traines) will help keep you injury free, and proper clothes will be much more comfy than an old t-shirt, especially in winter
Good luck!