Why is my iMessage marked 'Delivered' when it isn't?

A friend and I have been trying to fix a problem which came up with iMessage. I'm using an iPad mini (first generation, WiFi only) and she has an iPhone 5. We're both using iOS 8.3. For some reason, when I send iMessages to her, they are marked 'Delivered' even though she never received them. She can send me iMessages, no problem, but mine won't go through for her phone number or two emails associated with her Apple ID. We've tried turning data on and off as well as restarting, but it doesn't seem to change anything.

The problem seems to have started after we had a FaceTime, but I don't know if that's what caused it. After that, none of my messages went through. Also note that she can FaceTime me, but when I try it gets stuck on "Connecting".

Thanks for any help you brilliant people have!