The House has impeached (voted and noted as passed). It is the Senate’s role to try according to the constitution. If the paperwork is late for whatever reason, does not preclude the next step from happening, with or without the House’s cooperation. If the house was not ready, they should not have voted. That ended their role.
DK
Including dismissal without a trial.
Using the trial analogy, composing the indictment is not the end of the prosecutor's job. The prosecutor also has the role of presneting the charge to the judge, to the jury, something more than write the indictment, publish it in the paper, and file it in his own desk.
Anyway, the House role is not over. It is at the pivot point where it has resolved to exhibit the accusation to the senate. This is what they voted on, "we resolve to exhibit this to the senate." That passed. That exact language.