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Finally figured out how to delete my instagram account. I haven't touched it in ages but today I got email letting me know about my 23 "new" followers (some of which I know have been following the account for over a year) which provided some fresh motivation to just eliminate that. People can look at my cat pictures here instead.
This was surprisingly hard. I couldn't find a link to the delete your account page anywhere except in a help page found through an off-site search.