Sorry to say but it looks like you've cut into the bulb... but the good news is just keep doing what you're doing and it should be fine, just don't cut it anymore!

I keep mine in a warmer area rather than cool, but completely dry although still potted or with dirt around the tuber.

When you plant it in the spring instead of having a main shoot it will sprout from those side shoots you see. They will be smaller at first but there will be many more than you had last year.

Elephant Ears don't grow from a bulb like Tulips or Lilies. They actually grow from a tuber or rhizome and so are more like Dahlias or Cannas. You have cut into the tuber but it is okay so long as you leave a few of the roots and side stems which you have done. Actually, You can propagate more plants for elsewhere in your gardens simply by cutting the tuber into pieces. Just be sure to leave a few of the roots and side stems on each piece. Plant them as usual. They may be smaller the first year but will grow bigger each and every year thereafter. For more information search for "how to propagate an elephant ear" in Google - lots of sites come up. I hope this helps.