The easy one is why the screen stays the same without the battery connected: That's the way that e-ink works. Whatever is on the screen stays there until it is changed. There is no power used to maintain the screen.
Have you charged the battery? Don't assume it still has charge. And if everything else fails, leave it on charge overnight. It shouldn't work, but, a lot of people do this and find the device is running the next day.
What happens when you turn it on? Does the power light do anything? Does the screen light turn on?
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