Safety First!
Now that I got that out of the way, you are ABSOLUTELY liable for workplace injury. You might be able to fall back on gross negligence, but that's not likely. Your employee would have to basically been seen on camera hitting you when you ask her to put on PPE.
To put it another way, let's just say you're screen cleaner does get hurt while on the job, getting chemicals in their eyes for example. What would you do? Of course you would immediately drive them to the hospital, or call some kind of emergency service for them. At that point, are you going to leave them with the bill? I doubt it. So even if you don't get sued, you're still liable.
It's this kind of crap that makes my blood boil. Take care of the people who work for you and make sure they are as safe as you can make them. Employers who don't care screw it up for the rest of us and the darn government comes in and regulates everything after that!
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We require safety goggles and respirators (masks) when using screen cleaning chemicals and during several other potentially hazardous situations. It's not negotiable. Be an @$$hole about it if you have to. Better to be a jerk than to have someone get hurt 100% of the time.
We have a no-nonsense policy on safety. Any reasonable request for safety equipment is honored. If someone wants a $40 pair of eye protection, we'll get it - as long as they wear it. It's the one line item that there is no reasonable budget for. If it comes down to it, I'll take a pay cut that month.