That sounds pretty exotic for a server and I have no idea what jellyfin is but it doesn't seem to be a HTTP server.
See, you want to stream to the web aka http browser of the Wii U, so you need a http server to begin with. Then from there you can add plugins to that http server. Like HLS (HTTP live streaming) - Not sure the Wii U supports this but it's just one of many examples.
While these live streaming solutions might not be developed for your specific use case they at least have the functionality to transcode and stream to a web browser at the same time (that's the whole purpose of a live stream, right?). Also pretty sure many of them have a function to not stream a live picture but a video file (commercialy streamers need to switch seamlessly between live and pre-recorded). So while not directly produced for your usecase the software might still be able to do what you want.