The battery arrived Tuesday and yesterday I wanted to replace them yesterday. So I opened the Dremel and took a look inside...

Okay, that looks kind of easy. But then I found out the battery was soldered to the bottom copper plate. Darn...
I snapped it off the plates and then placed in the new battery. Closed the Stylus again and tried the power. Nothing. Darn...so I set it aside.
This morning I had the idea of connecting the two small strips on the top of the new battery, maybe that would help. Sparks! Sparks! Sparks! The battery also turned hot and one of the strips was fried off. Oops. Did I tell you I know jack about electricity...?
So as a final idea I simply turned the battery upside down (so with the fried strips to the bottom) and tried again. Bzzzzzzzzt...it works!
MWUAHAHAHA! It works! So although the battery is a bit fried (doh), the machine works again and very well too. I placed in the shed for now. Nobody home at the moment and don't want to find my home burned down when I get back...

So no, doing this myself wasn't the wisest idea, but I became wiser and fixed it in the end. And I'm thrilled the Stylus works again.