I used pasteurized egg white, not egg white powder.
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Assuming you iced a pretty standard sugar cookie, you should be good for three to five days. Any cookies that are not going to be eaten with hours of cooling/frosting, should be stored in an airtight tin or zip bags. You should also be able to freeze these for later consumption if you like.
Especially if you use meringue/egg white powder, you should definitely be good.