If you only have two wires at the switch box, then you don't have a neutral. Don't let the white wire fool you, you have a hot wire & a wire that is controlled by the switch.

Also, any receptacle you add to the bathroom needs to be Ground Fault Protected & the device you're proposing to install isn't.

Considering the multi-megawatt hairdryers on the market, I think your best bet would be to add a dedicated circuit the would feed a GFI receptacle. The alternative would be to replace the 2 wire cable with a 3 wire cable (could be hard to accomplish), enlarge the box to 2 gang & install the switch and a GFI receptacle.

Tom

the gfi is considered in some areas as a stop gap for non grounded circuits. but if you can run a new wire with ground to the panal, all the better.

but the swith may be wired in a loop to not allow this.

the power is actually at the light and then a black an white go to the switch and complete the loop. check for power at the light when the switch is off, to confirm this.

cs, Tom O said it all. No need to check for power at the light. There is no neutral in the switch box and Tom O's advice was 100% correct.

Ron