All the rooting methods for WP (and W10M) either require a custom ROM (definitely needs USB) or require installing "homebrew" apps that are not allowed in the store, and therefore must be "sideloaded".
Unfortunately, it's probably not possible to sideload apps on stock WP8.1. First, your phone would need to be "developer unlocked" to allow sideloading at all; this requires a USB connection. Next, the apps must be deployed to the phone; this is conventionally done via a USB connection. W10M fixes both of these problems - you can dev-unlock and sideload in W10M without ever using USB - but I don't think you can upgrade a 820 to W10M anymore without editing its registry, which requires a sideloaded app.
It may be possible to do the sideloading another way; at the very least "enterprise" (business) apps can be installed by just opening the app bundle file on the phone. However, marking an app as an "enterprise" app requires a special digital signature on it (using a key that I believe Microsoft has to provide) and also the phone needs to be associated with that "enterprise".
In any case, WP is basically a dead platform, even compared to W10M; Microsoft no longer supports it for most purposes, so I'm not even sure if you'd be able to dev-unlock it anymore. W10M phones can be had cheap, and are not only easy to root, you don't even need to root them in order to turn on device encryption. (Also, while rooting will definitely make it possible to turn on device encryption, I'm not sure if there's actually an app that does that for WP.)