Don't send videos
Any time I've had a video sent to me from my sister, it gets so compressed and grainy that you'd swear it was recorded on a phone from 2003 and not an iPhone5S.
If you want to share videos, my personal suggestion is Youtube or Facebook. With Youtube you can even set them to be private (only you can see them) or hidden (people need the link to see it -- they can't be found by searching or browsing your profile). They get the full quality video in this way as well. And, you have the ability to delete it later if you'd like.
That's rough it's been giving you trouble for so long. The best suggestion I can give you for the phone is to see if you can record a video that's 1 megabyte (MB) in size or less. smplyunprdctble has mentioned some alternatives that will allow to show the video without having to adjust the size.