Hi- at least you've started actually doing some running before deciding to do a marathon- that's a start! You will need a full year of progressive;y increasing your distances if you want to do yourself justice, but it is possible.
First thing to undertsand is that your chances on getting in on the ballot are very small, so , go ahead and enter, and start trainig, but be prepared that you will hear, in a few months time, that you have been unsuccessful. The best option is then to choose another spring marathon ( there are many) and keep trainig, there are lots of great races out there, and while you're at it, get off your treadmill, out into the big wide world, and start doing some local events. You can already do 5k- so find a park run, then a local 10k to enter, aim to be doing half marathons by the autumn, and the world's your oyster!
I started as a treadmill runner, and it's a good way to start, and great for speed trainig or bad weather trainig, but now it's the spring, - run outside, it's so much better!- it will take time to get accustomed to pacing yourself ,and feel very odd at first, but go for it, slowly, and soon you'll wonder why you ever preferred the tready.
If it MUST be london, you'll probably end up having to go down the charity route, in order to get a place.
Godd luck, join a newbie thread, get your self a good 5k>10k plan, and get out there!