To add to @Andrey, you should be able to achieve an accuracy of 98+% accuracy on the training with a simple 2 layer nn with 250-350 hidden nodes, sigmoid activation, learning rate of 0.1. I ran my test 1,000,000 iterations via stochastic gradient descent. However, I'm very sure it converged far earlier.
An example showing how I structured the experiment and results: herehere