My friend Zoe and I went to Spitalfields City Farm a few weeks ago with our kids, a 2 year old and a 9 month old. Really quick train ride to Whitechapel, and then a 5 minute walk from the station. My daughter loved wandering around the farm and looking at all the animals. We walked up Brick lane, grabbed some lunch by the Truman brewery, bought some bagels, and got the train back from Shoreditch High Street.
This was before the full service opened and it was easy enough, I'm sure it is much easier now. The lack of step free access was a bit of a pain with buggies at Whitechapel, but there were enough staff on hand to help out.
It was a very pleasant day out, we left at 1045 and were back home for nap time by 230pm.