It is simply a Republic according to our organic law: "The United States shall guarantee to every State in this Union a Republican Form of Government" Const IV.4
also, cf:
"... democracies have ever been spectacles of turbulence and contention; have ever been found incompatible with personal security or the rights of property; and have in general been as short in their lives as they have been violent in their deaths ... A republic, by which I mean a government in which the scheme of representation takes place, opens a different prospect, and promises the cure for which we are seeking ... great points of difference between a democracy and a republic ..." Federalist Paper 10
You missed out the qualifying statement before that:
From this view of the subject it may be concluded that a
pure democracy,
by which I mean a society consisting of a small number of citizens, who assemble and administer the government in person, can admit of no cure for the mischiefs of faction. A common passion or interest will, in almost every case, be felt by a majority of the whole; a communication and concert result from the form of government itself; and there is nothing to check the inducements to sacrifice the weaker party or an obnoxious individual...
A republic, by which I mean a government in which the scheme of representation takes place, opens a different prospect, and promises the cure for which we are seeking.
Let us examine the points in which it varies from pure democracy, and we shall comprehend both the nature of the cure and the efficacy which it must derive from the Union.
Federalist Paper 10
As I mentioned earlier, he is specifically talking about
pure democracy - direct Athenian democracy. There would be not need for the qualifier 'pure' if democracy didn't have a broader meaning.
His definition for Republic is that of a representative democracy, as contrasted with a pure democracy. There is no reason that a republic, in general, needs to be democratic, so when you say it is 'not a democratic republic' you are wrong. This is implicit in the definition given by Madison.
Also, the congress, as the constitution states is chosen by the people and has 'all legislate powers':
The Senate of the United States shall be composed of two Senators from each State,
elected by the people thereof
The House of Representatives shall be composed of Members
chosen every second Year by the People of the several States
"All legislative powers" come form the people via elected representatives, not 'organic law'. This is a democracy today, just as it was when the (very clearly democratic) Republic was founded.