How can I give my sister my iPad 2 with the videos and music from my iTunes account intact?
Once she decides to sync the device once, the contents will still be removed. Your sister won't be able to back up the videos and music or sync it to her iCloud, but it should keep playing all the media for months to come. If she erases it or syncs it with a new computer - she can return it to you for a reload or just start fresh with her songs at that point.

To remove your accounts, open the Settings App:


Remove the passcode (optional but nice to for her to choose one)
iCloud - tap Delete Account and delete all the content. Optionally delete your backup of this device in Storage &amp; Backup if you don't care to keep that backup around and free up your cloud space.
Mail, Contacts, Calendars - delete all accounts and data here
Notes, Reinders, Messages, Facetime - same as above
Safari - clear cookies and history
iTunes &amp; App Store - Sign out by tapping the account name for Apple ID
Music - Sign out of Home Sharing by tapping the Apple ID
Videos, Game Center - same as above


Sign out of social media - Twitter, Facebook, Flickr, Vimeo


Privacy - Location Services - Turn Off - that deletes your information from Apple if you chose to share it. Optionally review the apps that have requested access to private data and turn them off / delete the app so she can choose what she shares.
Privacy - Advertising - Reset Advertising Identifier and optionally Limit Ad Tracking


She should be fine adding her Apple ID and just will want to watch for prompts that indicate if she will ever wipe your videos and music. Depending on what you choose, you might just erase all content and settings and copy the Music and Videos to a $50 hard drive or large USB drive so she can incorporate them in her iTunes library rather than have you surgically remove things. The out of box experience does guide a new user to understand what accounts and settings she can opt in and out of.