Hi Jane,
I've done a lot of work (and authored a few guides) on the Switch itself, but I'm afraid I never got into the dock. But your problem reminded me of another person who had the same issue. He got it working partially but I never heard if he found a proper fix for it.
Anyway, it might be worth your while to read over his question and see if anything helps you with yours.
[post|788691|SOLVED: Nintendo switch is charging, but not outputting to a TV. - Nintendo Switch - iFixit|new_window=true]