How Long Do The Dems Have If Brown Wins?

Fifteen days:

Looking over these statutes, it seems clear that unless the result is very, very close (think Al Franken and Norm Coleman in Minnesota, or Scott Murphy and Jim Tedisco in NY-20), we should probably know on election night who has been elected when the vast majority of votes are counted. But even then, state law is clear that a certificate of election cannot be issued until at least 15 days later.

And if Senate Democrats insist on a completed certificate -- just as the Senate Dems did in their unsuccessful attempts to keep out Sen. Roland Burris (D-IL), and Senate Republicans did in their successful blocking of Al Franken during the Minnesota litigation -- that would keep the winner out for at least 15 days.