Absent an invalidation of the election results, Trump will no longer be president at noon on January 20th so there is no need to impeach.
If he remains in office, he will be the de facto office holder. We are in uncharted territory - but the fact remains that the Constitution will still be in play. The House and Senate will hold a vote to Impeach and Convict within a couple of days, and this will give them the legal and Constitutional cover to take whatever action they want. The Deep State will be unleashed, and it will either be Civil War or chaos.