The process is done by vote.
You are completely making up the part about a declaration from Nancy taking the place of a vote. That is not at all what is happening.
The idea would have been that whey would vote on whether the other vote is "deemed" to have passed. In either way the issue would have been dealt with a vote on a declaration (not just a declaration).
The "trick" (and it is a bit of a trick) is that the representatives instead of voting on an issue, they would be voting on voting on an issue. In either way the process involves a vote where a majority (i.e. 216 representative) is required to do.
This is why all the people whining when the Democrats discuss this procedure were perfectly fine with the procedure when the Republicans did it.
But this is now irrelevant. The bill will likely pass without this procedure. You will be happy. I will be happy. The Republic is saved. Republicans won't be put into concentration camps and the Constitution has survived another crisis.