Sorry I should have said, what you have done already is potentially dangerous (switch rating) and, in the UK probably illegal.
t
June 2008
Of course it will - switching off the light removes the power from it!
If you want it to stay on you need the type with an independent live feed (as well as the pull cord input) and a timer. Fitting one of these to a bathroom is a job for a qualified electrician.
howtomendit.com for free repair help, information and advice.
Try fixfault.com to fix lighting.