When you charge the battery, does the light change from red (or reddish-blueish) to blue? Are you using the cable from Kobo, or another cable? It shouldn't be draining as quickly as you say, so I wonder if it's actually charging properly?
Try using the cable from Kobo, just to eliminate any other issues. It needs to charge about 3 hours to get to the blue light.
If it's not lasting more than a day after that, I would contact Kobo and try to get a replacement.
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