Current branching in transformer circuit
From what I have learnt, since the transformer is ideal, the winding
  inductances of the transformer are infinite. Would not this however
  make the current going into the transformer 0 according to current
  branching?


The magnetizing inductance is infinite and the leakage inductances are zero: -



So, for a perfect ideal transformer Xm is infinite and all the other inductors are zero. The transformer symbol in the middle should not be regarded as anything but a perfect power transmission device (even though it is drawn as two coupled inductors.

Rc is infinite and all the other resistors are zero.

Therefore the current does flow into the transformer.