iPad consistently fails to deliver iMessages

Are you initiating a new contact, or appending to a historical one. If the latter try iniating the contact from the beginning and see if you have better luck.

Are all of the messages being sent to other Apple iOS devices?

Using FaceTime http://support.apple.com/kb/ht4319

Troubleshooting FaceTime http://support.apple.com/kb/TS3367

The Complete Guide to FaceTime + iMessage: Setup, Use, and Troubleshooting

http://tinyurl.com/a7odey8

Troubleshooting FaceTime and iMessage activation

http://support.apple.com/kb/TS4268

iOS: FaceTime is 'Unable to verify email because it is in use'

http://support.apple.com/kb/TS3510

Using FaceTime and iMessage behind a firewall

http://support.apple.com/kb/HT4245

iOS: About Messages

http://support.apple.com/kb/HT3529

Set up iMessage

http://www.apple.com/ca/ios/messages/

iOS and OS X: Link your phone number and Apple ID for use with FaceTime and iMessage

http://support.apple.com/kb/HT5538

How to Set Up & Use iMessage on iPhone, iPad, & iPod touch with iOS

http://osxdaily.com/2011/10/18/set-up-imessage-on-iphone-ipad-ipod-touch-with-io s-5/

Set Up Alert Sounds

http://www.quepublishing.com/articles/article.aspx?p=1873027&seqNum=3

Extra FaceTime IDs

http://tinyurl.com/k683gr4

Troubleshooting Messages

http://support.apple.com/kb/TS2755

Troubleshooting iMessage Issues: Some Useful Tips You Should Try

http://www.igeeksblog.com/troubleshooting-imessage-issues/

Setting Up Multiple iOS Devices for iMessage and Facetime

http://macmost.com/setting-up-multiple-ios-devices-for-messages-and-facetime.htm l

FaceTime and iMessage not accepting Apple ID password

http://www.ilounge.com/index.php/articles/comments/facetime-and-imessage-not-acc epting-apple-id-password/

Fix Can’t Sign Into FaceTime or iMessage iOS 7

http://ipadtutr.com/fix-login-facetime-imessage-ios-7/

FaceTime, Game Center, Messages: Troubleshooting sign in issues

http://support.apple.com/kb/TS3970

Unable to use FaceTime and iMessage with my apple ID

https://discussions.apple.com/thread/4649373?tstart=90

iOS 7 allows you to block phone numbers or e-mail addresses from contacting you via the Phone, FaceTime, or Messages

http://howto.cnet.com/8301-11310_39-57602643-285/you-can-block-people-from-conta cting-you-on-ios-7/

How to Block Someone on FaceTime

http://www.ehow.com/how_10033185_block-someone-facetime.html

My Facetime Doesn't Ring

https://discussions.apple.com/thread/4155326

How to watch FaceTime calls on the big screen with Apple TV

http://www.imore.com/daily-tip-ios-5-airplay-mirroring-facetime

Send an iMessage as a Text Message Instead with a Quick Tap & Hold

http://osxdaily.com/2012/11/18/send-imessage-as-text-message/

To send messages to non-Apple devices, check out the TextFree app https://itunes.apple.com/us/app/text-free-textfree-sms-real/id399355755?mt=8

How to Send SMS from iPad

http://www.iskysoft.com/apple-ipad/send-sms-from-ipad.html

How to Receive SMS Messages on an iPad

http://yourbusiness.azcentral.com/receive-sms-messages-ipad-16776.html

Apps for Texting http://appadvice.com/appguides/show/apps-for-texting

You can check the status of the FaceTime/iMessage servers at this link.

http://www.apple.com/support/systemstatus/

 Cheers, Tom 😉

Settings > Messages > Send & Receive > is the only one you really have to worry about

but your girlfriend should also check these settings on both her phone and the equivalent setting on her laptop.

My best guess from what you just said is that she has her phone set up with both her phone # and her email address, but her laptop set up with only the email address, and you are using her phone # as your contact for reaching her.

But the other alternative is simply that she changed her settings on her laptop since the first message in the historical contact you are using. In which case the link to the laptop is simply broken, and by initiating a new contact it will repair itself.