I think a lot more goes into "best run" than just team success. That's certainly the dominant factor, but other aspects also have to be considered.
How much has the team done for the community?
Have they grown hockey in their region if it's not already a staple there?
Do players and other staff enjoy being a part of the franchise?
What do the fans think (both die-hard and casual)?
It's also not a quick, sudden change - these types of things take years, sometimes decades to change and are very slow-moving.
Personally, Tampa fits all of these, at least as far as I can tell. They get my vote.
Toronto's headed in the right direction and will probably be there soon. I think Florida does a lot right, but their location relative to the population centers hurts them a bit. If Dallas can ever get any consistency with success on the ice, they should absolutely be in the conversation. Vegas had an incredible first year from top to bottom, but I'd like to see them maintain that before I consider them the best-run franchise.