Have a look at
The TV everywhere' package is not for a TV but rather a is a streaming service that allows you to watch BT TV channels on your computer, tablet or smartphone. It does not allow mirroring to a TV. So it a service that is currently provided when you select certain BT TV packages. Which TV package do you have currently and what broadband speed ?
BT do have an Extra (Youview) box option that can be added to many but not all packages subject also to some internet speed. More details here
Extra Box with BT TV on the go or Extra Box with BT TV Everywhere as I think it's alternative name as a package addon would not be available to you if you wanted to retain Sky Sports.
Hovever BT TV on the go (BT TV Everywhere ) also comes with the Entertainment Plus TV pack which I think is the current name for the one you have.
From other forum posts in the past people have self installed a second box - however BT policy on this seems to have changed over time ... in some scenarios where people already have 2 Youview boxes supplied by BT (ie they had previously upgraded their box ) and therefore did not need BT to supply a second box they were able to simply take the extra box subscription.
I suspect the desire to have Sky Sports may mean you will not opt to go down the extra box subscription route.