Will a drill bit that came with a dremel be good enought to drill out a screw?
It should be good enough, even though you are using a rue of thumb.  You might try a screw extractor.



Here's the instructions on using a screw extractor.

Here's a link from a previous post on suggestions on removing damaged screws.

If you extract it in a good way you won't damage anything and can reuse the old parts.  The screw extractor kit is handy to have around, even if it is only used a couple times a year, if that much.  It easily gets the bad screw out, get a replacement handle and screw a new set screw back in.  If you drill it out, the you risk damaging the threads or you might have to tap the hole and go with a large size that might stick out more than the designed screw.