You can only watch The Roku Channel itself on a non-Roku device. If any of the streaming services you use have an iOS app then you can install it and use that to watch. You'd need a separate app for each service unless that service is available via The Roku Channel, but to be honest I don't know how/if that works.
I know zero about anything Apple, but if there's an HDMI input adapter you can get for your iPad then you could possibly use it as a display for a Roku device.
Roku Community Streaming Expert
Help others find this answer and click "Accept as Solution." If you appreciate my answer, maybe give me a Kudo.