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This is my current (well, since about March) project. Lovely old Royal Enfield Lightweight Sports (the most hilarious misnomer by modern standards), around WWII era as far as I've managed to find out. I'm trying to restore it as originally as possible. Will probably make a little thread of its own for this because I have many questions, but for the moment here is a picture or two:

When I was riding it around a bit I took off the chainguard/mudguards because they were the most rattly irritating things in the world, couldn't be bothered to fix them back on just for a photo so they're just resting on there. Just to pre-empt anyone who points that out ;). Chain is also off because I've been doing some bits and bobs.

This is brilliant, difficult to make out at this size, but the tiny writing underneath the cannon says "Made like a gun." Class. (Royal Enfield made a lot of the munitions used in WWII).
So, I need to replace the BB, do some further cleaning & polishing, probably find a nice saddle of the right vintage, and replace the Sturmey Archer hub with a single speed or fixed (which I think is correct for the era, this is one of the things I will enquire about when I get around to doing some in-depth knowledge-begging).
I love this bike. They weren't lying when they said "Made like a gun," this is a seriously heavy beast, but once it's got going it's absolutely delightful to ride. And the rod brakes make me do a little happy squee of excitement. I realise I've gone on a bit, but borrowed a mate's camera to take some photos of it this afternoon and I've fallen in love all over again, hence gushing. Please do forgive :).
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Like BlueQuinn said, Greenway is a good option. From Stepney, you could go up Cambridge Heath Road, then through Victoria Park, and then join the Greenway at Hackney Wick. Follow the Greenway til it crosses the A11 (impossible to miss), and then you're only on that hideous road for a few minutes til you get to Stratford.
Or you could go nuts and head towards Devons Rd DLR on that long wiggly St Paul's Way/Devons Rd/Devas St road, then join Manor Road and get to Stratford via West Ham, but it seems a very long route, which I've never done, and looking at Google Maps there seem to be a few odd junctions.
I reckon the Vicky Park/Greenway is the best option, and after a while of regularly doing that stretch of the A11 she'll probably get used to it and won't think anything of it.