A tough question, as every situation is different. For me, I was like Saphie - my co-workers & boss already knew. I crashed and burned before admitting it to anyone. Once I came back from my time in detox (just a few days) everyone was very supportive and wonderful to me. I guess they were just relieved that it was out in the open and they didn't have to walk on eggs around me anymore.
I too thought the eyedrops, perfume & mouthwash were doing the trick - but a friend finally told me it was the change in my personality that was a dead giveaway. I think it depends on where you work and the type of company it is. I was a longterm employee at a hotel where we were like family, so I couldn't really hide it.
I hope you give it up, since it's causing you such misery and stress. Let us know how it's going.