Good questions.
Answers:
1. Eye-Fi is much slower than built-in wifi, and you'd lose the ability to trigger the camera. Built-in wifi is far superior to Eye-fi.
2. Upgrading from iPad Air 1 to a newer iPad will probably not help make it any faster, because the speed issue is probably the wifi throughput, not the processing speed. Newer iPads have better processors, but not better wifi (at least not when talking to your camera). So no, don't upgrade your iPad if you're only trying to fix this.
Here are some things to try. Make sure your image size is set on S, and for testing turn OFF RAW to see if that helps.
A) for testing, try connecting your ipad's wifi directly to your Canon 6D ("Camera mode" in canon terms). Does this make it faster? If so then your wifi router probably is too far away from the camera, or not very good.
B) for testing, try varying the distance between your camera and ipad (assuming you're still in camera mode wifi). They should be about 1 meter apart for optimum testing. Does this make it faster?
If you DO want to spend more money: the 6D is one of the worst wifi performers of all Canon cameras. A cheap Canon 750D is *much much* better! buy one of those? cheaper than an ipad probably
let me know what you find and i'll help.