I had this today and was able to fix.
To get the twist-off cover off, I applied heat directly to the keyboard near the cover. I used the side of heated oven, but a soldering iron work too. I used a nickle and pliers to unscrew the cover. (I mangle up a quarter first, a nickle worked better.)
To get the batteries out, I drilled a small hole in the stuck battery. Drilling it loosened the battery. Then I screwed a wood screw into the hold I drilled. I was able to pull out the battery by grabbing the wood screw with the pliers.
The second battery was loose, but gunk and plastic cover of battery was stuck to the inside of the keyboard. I scraped off the plastic, and used aluminum foil to scrub out the inside. Then I cleaned it with vinegar and a q-tip.
I put in new batteries and I was to use the keyboard to type this.
Now I plan on removing the batteries when I'm not using the keyboard. ;-)