"There must be at least one outlet in each bathroom, adjacent to the sink, in addition to any outlet that may be incorporated in the light fixture. All such outlets *must* be GFCI-protected. "The NEC says that switches may not be installed inside bathtubs or showers. The CEC says that switches may not be installed "within reach" of bathtubs or showers (consult an inspector if you can't make it at least four feet)." so it says at :
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the missing puzzle pieces: does the light switch have a separate additional switch on it? how many wires are in the light's box? where are all these devices?
like in our yellow bathroom, if the switch and the outlet are in the same small single sized box, get some plastic wiremold parts including a deep surface mounted double box and adapter to your single and put a gfi side by side with the light switch. choose the lighted handle modern rocker switch easily found in the dark.
this project may also involve an electrician or maybe a nice new lighted medicine cabinet. it requires a gfi which is better than what you presently are describing, and gfi's hook right up to those two outlet wires just fine even if there was a bad ground. so at least put a gfi in to replace the outlet.
at a different house we installed new flourescent lights with the gfi and now there are 2 lights for shaving and a switched outlet and an unswitched one. when you get done plugging in all the rechargable shavers and trimmers and hair dryers you'll want a power strip in the bathroom.