Electronic circuit analysis: Find V_out with several MOSFETs
If there was no battery, it would just flow to the right, 


In this kind of circuit, the current source is not really meant to be an ideal current source. If there was no battery, the current source would be turned off too.


  So my way of solving it seems to be incorrect, but how else to I tackle the problem?


The circuit is a current mirror. If you research, you will find numerous explanations of how it works and how to calculate the currents through M2 and M3.