Alisha ( @flannelist ) has been working diligently on troubleshooting guides, and I'd suggest taking a look at the No Video Out guide they've written as the place to start.
Nintendo Switch No Video Out - iFixit Troubleshooting
Other than that, it appears you're not the only person with that problem. Ethan ( @nahdoesntexist ) was having the same issue, but unfortunately did not find a solution.
SOLVED: Nintendo switch is charging, but not outputting to a TV. - Nintendo Switch - iFixit
An interesting find here was that if you take the dock connector out of the dock and plug it into the Switch it works, but once it's installed in its plastic shell it no longer works. I'm not sure if that's of any help or not.